advances in internal model principle control theory
TRANSCRIPT
Western University Western University
Scholarship@Western Scholarship@Western
Electronic Thesis and Dissertation Repository
2-11-2011 12:00 AM
Advances In Internal Model Principle Control Theory Advances In Internal Model Principle Control Theory
Jin Lu The University of Western Ontario
Supervisor
Dr. Lyndon J. Brown
The University of Western Ontario
Graduate Program in Electrical and Computer Engineering
A thesis submitted in partial fulfillment of the requirements for the degree in Doctor of
Philosophy
© Jin Lu 2011
Follow this and additional works at: https://ir.lib.uwo.ca/etd
Part of the Controls and Control Theory Commons
Recommended Citation Recommended Citation Lu, Jin, "Advances In Internal Model Principle Control Theory" (2011). Electronic Thesis and Dissertation Repository. 90. https://ir.lib.uwo.ca/etd/90
This Dissertation/Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected].
ADVANCES IN INTERNAL MODELPRINCIPLE CONTROL THEORY
(Spine title: Advances In Internal Model Principle Control Theory)
(Thesis format: Integrated Article)
by
Jin Lu
Graduate Programin
Engineering ScienceElectrical and Computer Engineering
A thesis submitted in partial fulfillmentof the requirements for the degree of
Doctor of Philosophy
The School of Graduate and Postdoctoral StudiesThe University of Western Ontario
London, Ontario, Canada
c© Jin Lu 2011
THE UNIVERSITY OF WESTERN ONTARIO
School of Graduate and Postdoctoral Studies
CERTIFICATE OF EXAMINATION
Supervisor: Examiners:
Dr. Brown, Lyndon J. Dr. Chen, Xiang
Supervisory Committee: Dr. Kermani, Mehrdad
Dr. Jiang, Jin Dr. Rohani, Sohrab
Dr. McIsaac, Ken Dr. Sidhu, Tarlochan
The thesis by
Jin Lu
entitled:
Advances In Internal Model Principle Control Theory
is accepted in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Date Chair of the Thesis Examination Board
ii
Abstract
In this thesis, two advanced implementations of the internal model principle
(IMP) are presented. The first is the identification of exponentially damped sinu-
soidal (EDS) signals with unknown parameters which are widely used to model audio
signals. This application is developed in discrete time as a signal processing problem.
An IMP based adaptive algorithm is developed for estimating two EDS parameters,
the damping factor and frequency. The stability and convergence of this adaptive
algorithm is analyzed based on a discrete time two time scale averaging theory. Sim-
ulation results demonstrate the identification performance of the proposed algorithm
and verify its stability.
The second advanced implementation of the IMP control theory is the rejection
of disturbances consisting of both predictable and unpredictable components. An
IMP controller is used for rejecting predictable disturbances. But the phase lag
introduced by the IMP controller limits the rejection capability of the wideband
disturbance controller, which is used for attenuating unpredictable disturbance, such
as white noise. A combination of open and closed-loop control strategy is presented.
In closed-loop control mode, both controllers are active. Once the tracking error is
insignificant, the input to the IMP controller is disconnected while its output control
action is maintained. In the open loop control mode, the wideband disturbance
controller is made more aggressive for attenuating white noise. Depending on the
level of the tracking error, the input to the IMP controller is connected intermittently.
iii
Thus the system switches between open and closed-loop control modes.
A state feedback controller is designed as the wideband disturbance controller
in this application. Two types of predictable disturbances are considered, constant
and periodic. For a constant disturbance, an integral controller, the simplest IMP
controller, is used. For a periodic disturbance with unknown frequencies, adaptive
IMP controllers are used to estimate the frequencies before cancelling the distur-
bances. An extended multiple Lyapunov functions (MLF) theorem is developed for
the stability analysis of this intermittent control strategy. Simulation results justify
the optimal rejection performance of this switched control by comparing with two
other traditional controllers.
Keywords: Internal model principle; Exponentially damped sinusoid; Inter-
mittent control; Disturbance cancellation; Switched systems; Multiple Lyapunov
functions
iv
Co-Authorship Statement
Some of the research work in this thesis, was summarized into several technical
papers published or submitted for publication in conference proceedings or IEEE
journal, co-authored by Jin Lu and Dr. Lyndon J. Brown.
Chapter 2 has been published in the Proceedings of the 17th IFAC World
Congress. Chapter 3 has been submitted to the IEEE Transactions on Automatic
Control. Part of Chapter 4 has been submitted to the 24th Canadian Conference on
Electrical and Computer Engineering.
The stability theorems and their proofs presented in this thesis were developed
by Jin Lu and reviewed by Dr. Lyndon J. Brown. The simulations and tests in this
thesis, including simulation runs and results analysis, were conducted by Jin Lu. The
papers and the whole thesis were written by Jin Lu and reviewed by Dr. Lyndon J.
Brown.
v
Acknowledgements
I would first like to express my sincere appreciation to my supervisor, Dr.
Lyndon J. Brown, for his guidance, patience and encouragement throughout the past
few years of my study at Western. It has been an honour working with such an
intelligent and gracious mentor.
I would also like to thank my friends here in London for their support and help
throughout the duration of my study at Western. It is a memorable experience and
I wish them all the best in the future.
My final expression of gratitude is to my family. Their unconditional support
has been an invaluable asset not only in this research but also in every aspect of my
life.
vi
Table of Contents
CERTIFICATE OF EXAMINATION . . . . . . . . . . . . . . . . . . . ii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Co-Authorship Statement . . . . . . . . . . . . . . . . . . . . . . . . . . v
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Frequency estimation . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Exponentially damped sinusoidal signals . . . . . . . . . . . . 13
1.2.3 Switched control systems . . . . . . . . . . . . . . . . . . . . . 17
1.3 Introduction of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 28
1.3.1 Adaptive internal model principle control algorithm . . . . . . 28
1.3.2 Intermittent cancellation control algorithm . . . . . . . . . . . 31
1.4 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 32
1.5 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 34
vii
2 Identification of Exponentially Damped Sinusoidal Signals . . . . 43
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 Internal Model Principle Based Adaptive Algorithm . . . . . . . . . . 46
2.2.1 Adaptive algorithm in continuous time . . . . . . . . . . . . . 46
2.2.2 Derivation of the adaptive algorithm in discrete time . . . . . 49
2.3 Convergence and Stability Analysis . . . . . . . . . . . . . . . . . . . 52
2.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3 Combining Open and Closed-Loop Control Via Intermittent Integral
Control Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2 Intermittent Integral Control System . . . . . . . . . . . . . . . . . . 73
3.3 Multiple Lyapunov Functions Based Stability Theory . . . . . . . . . 76
3.4 Switched System Model and Stability Theorem . . . . . . . . . . . . 79
3.5 A Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.5.1 Controller 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.5.2 Controller 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.5.3 Intermittent integral controller . . . . . . . . . . . . . . . . . . 86
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Appendix A Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . 92
Appendix B Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . 95
viii
4 A Combination of Open and Closed-loop Control for Disturbance
Rejection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.2 Open and Closed-Loop Control Strategy . . . . . . . . . . . . . . . . 111
4.3 Switched System Model and Stability Analysis . . . . . . . . . . . . . 115
4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.4.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.4.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.2 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
ix
List of Tables
3.1 Definition of parameters for intermittent integral control . . . . . . . 74
3.2 Guidelines for parameter selections . . . . . . . . . . . . . . . . . . . 75
3.3 Cost function values of controller 2 & controller 3’s step responses . . 88
4.1 Guidelines for parameter selections . . . . . . . . . . . . . . . . . . . 115
4.2 Costs of disturbance rejection performance with frequency adaptation 124
4.3 Costs of disturbance rejection performance without frequency adaptation124
4.4 Costs of rejection of multiple sinusoids . . . . . . . . . . . . . . . . . 132
x
List of Figures
2.1 Block diagram of an internal model principle control system . . . . . 47
2.2 Block diagram of the adaptive IMP control system . . . . . . . . . . 51
2.3 Block diagram of an adaptive IMP feedback control system for multi-
EDS signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.4 Single EDS signal with a parameter step change and error response . 60
2.5 Parameter estimation of a single EDS signal with a parameter step
change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.6 Single EDS signal with time-varying damping factor and error response 62
2.7 Parameter estimation of a single EDS signal with time-varying damp-
ing factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.8 Multi-EDS signal and error response . . . . . . . . . . . . . . . . . . 64
2.9 Parameter estimation for EDS mode d1(k) . . . . . . . . . . . . . . . 64
2.10 Parameter estimation for EDS mode d2(k) . . . . . . . . . . . . . . . 65
3.1 Block diagram of an intermittent integral control system . . . . . . . 74
3.2 Integral gain of the intermittent integral control system . . . . . . . . 89
3.3 Tracking errors and control signals of the three control systems . . . . 90
3.4 Comparison of steady state control signals of controller 2 & controller 3 91
3.5 Multiple Lyapunov functions stability . . . . . . . . . . . . . . . . . . 93
3.6 Switched system with different equilibria . . . . . . . . . . . . . . . . 95
4.1 Block diagram of open and closed-loop control system . . . . . . . . . 112
4.2 Intermittent IMP control and tracking error . . . . . . . . . . . . . . 125
4.3 Frequency estimation of the intermittent and non-intermittent control 126
xi
4.4 Tracking error during transient periods . . . . . . . . . . . . . . . . . 127
4.5 Intermittent IMP control action . . . . . . . . . . . . . . . . . . . . . 133
4.6 Frequency estimation using adaptive IMP algorithm . . . . . . . . . . 134
4.7 Transient tracking error comparison between intermittent and non-
intermittent control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
xii
Abbreviations
AFC Adaptive Feedforward Control
ANF Adaptive Notch Filter
CLF Common Lyapunov Function
CQLF Common Quadratic Lyapunov Function
EDS Exponentially Damped Sinusoid
EKF Extended Kalman Filter
FT Fourier Transform
HMM Hidden Markov Model
HR High Resolution
IMP Internal Model Principle
IF Instantaneous Frequency
LMI Linear Matrix Inequality
LQR Linear Quadratic Regulator
ML Maximum Likelihood
MLF Multiple Lyapunov Function
PLLF Piecewise Linear Lyapunov Function
STFT Short Time Fourier Transform
SVD Singular Value Decomposition
TFR Time Frequency Representation
xiii
1
Chapter 1
Introduction
1.1 Motivation
A classical problem in control theory is that of having the system output asymptoti-
cally tracking prescribed references and/or rejecting disturbances despite uncertainty
in plant and controllers, or so called zero-error output regulation problem [1]. In gen-
eral, disturbance signals can be classified into predictable and unpredictable signals.
The objective of this thesis is to develop control strategies that can achieve zero-error
regulation for certain types of exogenous inputs (reference or disturbance signals).
Predictable signals are also called narrowband signals whose essential spectral
content is limited to regions of narrow bandwidths. A typical class of narrowband
signals are periodic or quasi-periodic signals, which can be seen in many diverse ap-
plications, such as rotating machineries, computer disk drives, and power systems.
These signals are modelled as sinusoidal or sum of sinusoidal signals. The amplitudes
of the sinusoids in these models are constant. A natural extension of these signals
are signals whose amplitudes vary slowly, relative to the signal frequency. However,
in audio signal processing, music or speech signals are normally modelled as sinusoids
2
with faster time-varying amplitudes, particularly with exponentially damped ampli-
tudes. This type of sinusoid is called Exponentially Damped Sinusoid (EDS) with
the following representation in discrete time,
s(k) = aeσk sin(ωk + φ), k = 0, 1, 2, · · · (1.1)
where σ is the damping factor, ω is the frequency, a is the initial amplitude, and φ
is the initial phase of the signal. Both frequency ω and damping factor σ can be
time-varying, as long as these variations are small with respect to the value of ω.
Thus the amplitude of an EDS signal does not necessarily decrease as time evolves.
Depending on the value of σ, which could be positive or negative constant or time-
varying, the amplitude evolves differently. The signal form in (1.1) can be seen as a
generalized representation of pure sinusoidal or exponential signals by letting σ or ω
be zero respectively. If both parameters are zero, it is just a constant.
Different from predictable signals, unpredictable signals, such as white noise and
colored noise, have their spectral contents spread over a wide bandwidth. So they are
also called wideband signals. No controller can cancel unknown or uncertain wideband
signals. They can only be attenuated. But a uncertain narrowband signal can be
perfectly rejected or tracked by properly designed controllers. The perfect rejection or
tracking cannot be achieved without a perfect identification of the narrowband signal.
Thus there is a strong connection between zero-error output regulation problem and
signal identification problem.
3
When designing controllers to reject a narrowband disturbance, it is required to
have them providing closed-loop stability and output regulation. A plant-controller
combination is called structurally stable if these two requirements are satisfied when
certain system parameters are perturbed [1]. In order to achieve structural stabil-
ity, the plant-controller combination must utilize feedback of the regulated variable,
typically the tracking error between the plant output and the exogenous input, and
incorporate in the feedback path a suitably reduplicated model of the dynamic struc-
ture of the exogenous input signal. This is the main idea of the so called Internal
Model Principle (IMP), proposed by Francis and Wonham in 1976 [1]. The replicated
model of the exogenous input is called an internal model.
One of the simplest IMP controllers is integral controller, which is commonly
used for coping with constant signals. It is well known that zero tracking error can
be achieved using integral control for a constant input. IMP control has also been
applied to cancelling sinusoidal disturbances, due to their wide spread existence in
many applications. In this thesis, the identification of EDS signals with unknown
parameters is studied based on the IMP approach.
When a system is subjected to a disturbance with both predictable and un-
predictable components, an IMP controller and a wideband disturbance controller
can be implemented simultaneously for disturbance rejection. However, the rejection
capability of the wideband disturbance controller will be limited due to the phase lag
introduced by the IMP controller. Furthermore, if the predictable disturbance con-
4
tains multiple sinusoidal modes with unknown frequencies, especially if the number
of modes is large, it will be extremely difficult to tune these controllers. In order to
achieve an optimum rejection performance, the regulation variable can be connected
to the IMP controller intermittently. This intermittent control strategy also makes
the tuning feasible for multiple IMP controllers. This control idea was proposed by
Brown et al. in [2, 3]. The intermittent control system can be modelled as a switched
system. The stability analysis of the intermittent control strategy is presented in
this thesis, as well as an extension to the control strategy for dealing with different
predictable signals.
1.2 Literature Review
According to the internal model principle, when a replicated internal model of the
exogenous input is connected in the feedback path, the closed-loop system transfer
function, between the input and the tracking error, has a set of transmission zeros that
includes all the eigenvalues of the autonomous dynamical system which generates all
such input signal. Thus the exogenous input is asymptotically blocked in its transit
through the closed-loop system as a result of pole-zero cancellation. Specifically for
a sinusoidal signal with frequency ωc, it has a pair of critically stable poles at ±jωc
on the imaginary axis in the frequency domain. If the frequency ωc is known, an
accurate internal model with fixed parameters of the sinusoid can be constructed to
achieve zero-error regulation.
5
However, in many applications, the sinusoid’s frequency may vary over time or
is completely unknown. An IMP controller with fixed parameters will cause unaccept-
able tracking error. This is the main limitation of the IMP approach which requires
the knowledge of the reference or disturbance signals a priori, so that correspond-
ing internal model can be suitably produced. The accuracy of regulation depends
critically on the fidelity of the internal model. Even errors of less than one percent
in model parameters can lead to unacceptable residue errors. Therefore, accurately
estimating the periodic signal’s frequency becomes the most crucial task in regulation
problems.
1.2.1 Frequency estimation
Fourier transform (FT) is a standard tool for spectral analysis in signal processing
area. Although the FT is valid under extremely general conditions, there are some
crucial restrictions of Fourier analysis: the system must be linear and the data must be
strictly periodic or stationary. Natural phenomena measurements are essentially non-
linear and non-stationary. Alternatives to the FT for non-stationary signals include
the Hilbert transform and wavelet transforms. The Hilbert transform of a signal X(t)
is defined as Y (t) = Pπ
∫∞−∞
X(τ)t−τ dτ , where P indicates the Cauchy principal value.
The instantaneous frequency (IF) is then defined as ω(t) = ddt∠
(X(t) + jY (t)
)[4].
For narrowband signals, this definition matches our common sense idea of frequency,
however, it is not physically meaningful for non narrowband signals.
6
1.2.1.1 Time-frequency representation based methods
One type of the IF estimation techniques is based on the time-frequency represen-
tation. A time-frequency representation (TFR) is a signal representation in which
time and frequency information are displayed jointly on a 2-D plane. This represen-
tation is useful for analyzing signals with both time and frequency variations. The
short-time Fourier transform (STFT) and the Wigner distribution (WD) [5] are two
popular choices for TFR-based IF estimation. The STFT guarantees positivity and
is computationally efficient and very robust against noise. However, it suffers poor
time-frequency resolution. Although the WD has many desirable properties such
as high signal concentration in time-frequency, it suffers large cross terms between
multiple signal components in time-frequency, which makes it difficult to interpret
the WD in many practical applications. In [6], a hidden Markov models (HMMs)
based algorithm is used to track the peak of the STFT. The outputs of the HMM
tracker provides an estimate of the mean signal frequency as a function of time. In
this method, the states of the HMM are chosen to lie in some finite scalar set and
transition probabilities between elements of this set are known. White [7] generalized
the HMM formalism to include the case when the state set may be regarded as the
Cartesian product of a finite number of elementary state sets. This so-called Cartesian
HMM (CHMM) based method can track the IF and phase simultaneously. However,
the CHMM introduces a large number of states, which makes the computational cost
very expensive.
7
Kwok and Jones [8] proposed an adaptive short-time Fourier transform (ASTFT)
based instantaneous frequency estimation that inherits the advantages of the STFT
without most of its drawbacks. This is done by using different windows at each time
instance to achieve a good TFR. The adaptation rule is a generalized likelihood ratio
test based on the STFT. In addition, a post-ASTFT peak tracking algorithm further
improves the performance by following the continuous ridge in the time-frequency
plane and removing the spurious deviations. The algorithm is constructed under
a statistical detection and estimation framework and is an approximate maximum-
likelihood sequence estimator (MLSE) of IF tracks.
1.2.1.2 Filter based methods
Another type of IF estimation techniques are filter based, which includes extended
Kalman filter (EKF) frequency estimation, adaptive notch filter, and adaptive feed-
forward control. The EKF frequency estimator design proceeds from a state space
signal model of the process to be estimated. The signal model dynamics describe a
mechanism for how the process may be evolving. The initial stage in the filter de-
sign process is to perform system identification. Once the system has been identified,
Kalman filter theory endeavours to construct an optimal estimator for the state, given
the noise covariances Q and R, where optimality is measured in terms of covariance.
The EKF is derived by linearizing the signal model about the current predicted state
estimate and then using the Kalman filter on this linearized system to calculate a
8
gain matrix. This gain matrix, along with the nonlinear signal model and new signal
measurement, is used to produce the filtered state estimate and then an estimate of
the state at the next time instant. In developing extended Kalman filter based on sig-
nal models, the design compromises are to balance filter divergence and sensitivity to
noise [9]. However, the approach developed in [9] is characterized by a vector of three
design parameters. The tuning of these three parameters still remains a difficult task
due to the unclear cross-relationships between such parameters. Bittanti and Savaresi
[10] proposed a parametrization of the extended Kalman filter frequency tracker that
is characterized by just one parameter. This simplification allows an easier and more
transparent tuning of its tracking behaviour.
Adaptive notch filter (ANF) is a well-studied technique for removing or retriev-
ing sinusoids of unknown frequency from additive noise. Classically, the adaptive
notch filter is parametrized with the polynomial coefficients of its transfer function.
These coefficients are a function of the notching frequencies for which the notch filter
has or nearly has a zero gain. The frequencies are then computed from the estimated
transfer function coefficients. These techniques require stability monitoring, that is,
the model stability has to be checked after each adaptation, which leads to a lot of
additional computation [11]. In [12], the lattice structure was used to overcome the
stability monitoring problem for the general adaptive IIR filtering problem. This
lattice-based adaptive IIR notch filter features independent tuning of the notch fre-
quency and attenuation bandwidth. As opposed to minimizing an output error cost
9
function, this algorithm is designed instead to achieve a stable associated differential
equation. This results in a globally convergent unbiased frequency estimator in the
single sinusoidal case, independent of the notch filter bandwidth. Using a second-
order structure in the multiple sinusoids case, unbiased estimation of one of the input
frequencies is achieved by thinning the notch bandwidth. Dragosevic and Stankovic
[13] derived an expression for the notch filter output power to serve as a prerequisite
for finding the asymptotically optimal values of both pole contraction and forgetting
factors. The derived optimality conditions depend on a priori knowledge of the signal
and noise parameters.
Adaptive feedforward control (AFC) is an approach based on the phase-locked
loop technique commonly used in frequency-modulation communication systems. Bod-
son and Douglas [14] presented two algorithms for the rejection of sinusoidal distur-
bances with unknown frequency. The first is an indirect adaptive algorithm where the
frequency of the disturbance is estimated independently of the cancellation scheme by
an adaptive notch filter. The estimate is then used in another adaptive algorithm that
adjusts the magnitude and phase of the input needed to cancel the effect of the distur-
bance. The second is a direct adaptive algorithm in which a single error signal is used
to update the frequency and the magnitude estimates simultaneously. This algorithm
consists in extending the AFC scheme by integrating a phase-locked loop so that dis-
turbances with unknown frequencies can be directly cancelled. However, the gain of
the transfer function of the plant at the identified frequency is explicitly required in
10
order to implement this direct adaptive algorithm. Thus this algorithm could be very
complicated to implement in practice where the plants are often complicated. In [15],
a frequency estimator based on a magnitude/phase-locked loop approach is presented.
The estimator has the following features: simultaneous estimation of the frequencies,
magnitudes, and phases of the components of a periodic signal, simplicity in design
and implementation, and fast estimation/tracking of time-varying parameters. This
algorithm is derived from the direct adaptive algorithm presented in [14], with an ad-
justment by adding a proportional term Kf in phase estimation to improve frequency
tracking.
1.2.1.3 Signal subspace based methods
Estimating a set of parameters from measurements of the received signals is a prob-
lem of significance in many signal processing applications, such as direction-of-arrival
(DOA) estimation, system identification, and time series analysis. High-resolution
frequency estimation is important in numerous applications, which include the design
and control of robots and large flexible space structures. There have been several
approaches to such problems including maximum likelihood (ML) method and maxi-
mum entropy (ME) method. Although often successful and widely used, these meth-
ods have certain fundamental limitations due to the use of an incorrect model of the
measurements [16]. The MUSIC (MUltiple SIgnal Classification) algorithm was devel-
oped in the late 1970’s by Schmidt and Bienvenu independently [16]. This algorithm
11
first estimates the signal subspace from the array measurements. The parameters of
interest are then estimated from the intersections between the array manifold and
this estimated subspace. However, although the performance advantages of MUSIC
are substantial, they are achieved at a considerable cost in computation and stor-
age. Roy and Kailath [16] proposed a so-called estimation of signal parameters via
rotational invariance techniques (ESPRIT) approach that can be applied to a wide
variety of problems including accurate detection and estimation of sinusoids in noise.
It exploits an underlying rotational invariance among signal subspaces induced by an
array of sensors with a translational invariance structure, and generates estimates that
are asymptotically unbiased and efficient. This approach produces signal parameter
estimates based only on eigen-decompositions. It has important implementational
advantages over MUSIC in direction finding applications, as a result of reduction of
computation and storage costs. ESPRIT is also manifestly more robust with respect
to array imperfection than previous techniques including MUSIC [17].
1.2.1.4 Internal model principle based methods
In order to remove the main limitation of the IMP approach, which requires the
knowledge of the exogenous signals a priori, many approaches have been developed.
One of these developments is the repetitive control scheme presented by Tsao et al.
in [18]. Their period identification algorithm is based on the gradient minimization
of a quadratic energy function. They adopted a sampled data recursive scheme for
12
identifying the period of a periodic signal with a resolution finer than the sampling
interval. The fine adaptation of the controller sampling interval makes the identified
signal period an exact integer multiple of the controller sampling interval and renders
a superior tracking performance than that of the conventional fixed sampling interval
repetitive controllers. But their algorithm is often unstable at the starting phase of
identification, a low-pass filter is needed to stabilize their algorithm. Another disad-
vantage of their algorithm is that the speed of the convergence is slow. If the initial
condition and the basic period are not close enough, the adaptation may converge to
a local minimum at an integer multiple of the basic period.
Serrani et al. [19] employed a canonical parametrization of the internal model to
solve the problem of output regulation for nonlinear minimum phase systems driven
by an exosystem with unknown frequencies within known bounds. By adaptively
tuning the internal model, robust regulation with a semi-global domain of convergence
can be obtained. This approach adopts the conventional parallel connection of a
robust stabilizer and an internal model. Ding [20] introduced a parameter-dependent
state observer without following the parallel structure. In the state observer, a new
formulation of internal model is used to generate the contribution of the desired
input compensation to a state variable, which is then used in the control design. As
a result, the state estimates are involved with unknown parameters which appeared
in the internal model. The unknown parameters are dealt with in the control design
by adaptive backstepping technique. With this control design, the stabilisation and
13
compensation control efforts are combined together to achieve global stabilisation
and disturbance suppression with respect to state variables. Marino and Tomei [21]
designed an output feedback regulator which tunes its own internal model of a linear
stable exosystem with unknown frequencies and known order. For a stabilizable and
detectable linear system, this output regulator guarantees exponential convergence
of regulation error for any initial condition. In [22], the use of IMP is explored for
the rejection of time-varying of narrowband disturbances. In this control scheme,
the Youla-Kucera parametrization (Q-parametrization) is applied to the controller,
which makes it possible to insert and adjust the internal model in the controller by
adjusting the parameters of the Q polynomial.
1.2.2 Exponentially damped sinusoidal signals
The sinusoidal signal has proven its efficiency in modelling harmonic or quasi-harmonic
signals that present slow time variations. However, such modelling provides poor per-
formance when representing transient signals, typically signal onsets or fadings, which
are, by nature, localized both in time and frequency. It can be beneficial to produce
a signal representation based on waveforms allowing a better modelling of fast time-
varying signals [23]. Exponentially damped sinusoidal (EDS) signal is a more powerful
representation of an audio signal such as speech or music than the basic sinusoidal
signal.
Traditionally, the EDS signal is associated with a high resolution (HR) parame-
14
ter estimation method, such as matrix pencil, ESPRIT or Kung’s algorithm [24]. Hua
and Sarkar [25] presented an approach to exploit the structure of a matrix pencil of
the EDS signal for estimating the signal parameters. The SINTRACK method [26]
uses a matrix pencil algorithm for the detection and initial estimation of EDS signal
parameters. An adaptive Least Mean Square (LMS) algorithm is then used to track
the minor fluctuations in parameter values. The use of these two algorithms offers a
good compromise between accuracy and computation cost.
In [27], a subspace-based high resolution method is presented for the estima-
tion and tracking of the EDS signal parameters, the damping factor and frequency.
The estimation of the signal parameters is achieved in two steps: first the damping
factor and frequency are computed using a high resolution method, then they are
used to re-synthesis the signal. A simple re-synthesis method is able to realize the
adaptive tracking of the slow variation of the signal parameters. Gunnarsson and
Gu [28] proposed an analysis and synthesis system for music signals consist of mul-
tiple EDS components. The music signal is first divided into overlapped blocks with
fixed size which are approximately stationary. A least-squares ESPRIT method is
then applied to each data block for estimating associated signal parameters, includ-
ing frequency, damping factor, amplitude and initial phase of each EDS component.
The estimated EDS components are then tracked along the time direction under a
continuity constraint to the components in the music signal.
These subspace decomposition based HR methods are very efficient in the con-
15
text of audio modelling. One of their advantages is that they are not bounded by
the Fourier resolution limitation. This advantage turns out to be particularly con-
sistent whenever two spectral components of a complex sound are separated by a
distance smaller than the frequency resolution. Also, when there is frequency sliding,
these methods can be efficient. Another advantage of HR methods is their ability
to provide a good estimation of the parameters over a low number of samples. The
use of very short-time analysis windows is then possible without any important loss
of estimation performance which allows a good time resolution [23]. However, other
than their high computational complexity, these methods do not exploit the harmonic
relations between the frequencies and have a high complexity orders. Consequently,
these methods become ineffective when the length of the analysis segment is large
[24].
Other techniques for EDS parameter estimation include maximum likelihood
(ML) [29], polynomial or linear prediction methods, and higher order statistics based
approach [30]. Kumaresan and Tufts (KT) [31] presented a linear prediction method
by using the principal eigenvectors of the data matrix to separate signal subspace
from noise subspace in the form of singular value decomposition (SVD) and the
backward linear prediction equations of the noisy observations of the EDS signal.
When the analyzed signal is contaminated with a high level noise, this method exhibits
a superior performance. Based on the knowledge of the third or fourth order statistics
of the observed EDS signal, Papadopoulos and Nikias [30] extended this minimum
16
norm principal eigenvectors method (KT method) to higher order statistics domains.
The signal parameters are calculated by polynomial rooting of a vector of coefficients,
which is a solution of a linear system of equations involving third or fourth order
statistics.
All methods mentioned above have been proven to be conditionally efficient
in estimating parameters of EDS signals form a batch of measurement data. But
they all suffer from a common disadvantage, i.e., these techniques are all processed
off-line and therefore are not suitable for detecting and tracking the time-varying
parameters such as damping factors and frequencies. Zhang et al. [32] proposed a
novel method of estimating parameters of EDS signals by combining Hankel singular
value decomposition (HSVD) with extended complex Kalman filter (ECKF). The
HSVD algorithm is essentially used to obtain accurate initial state estimates from
a small number of samples. These estimates are subsequently being used by ECKF
which is capable of estimating EDS parameters and effectively tracking parameter
variations.
The EDS signal described in (1.1) is real. It can also be described in terms of
complex exponential signals. Let α = 12ae
jφ, and z = eσ+jω, we have
s(k) = αzk − α∗(z∗)k (1.2)
where the asterisk (∗) denotes complex conjugate. Among the methods discussed
above, except the iterative method presented in [24], all other methods use the com-
17
plex exponential description (1.2) to estimate the signal parameters. Since the fre-
quency can be identified by taking derivative of the phase of the complex exponentials,
it is easier to deal with complex signals than real signals.
1.2.3 Switched control systems
Many dynamical systems from various areas involve the interactions of discrete events
and continuous dynamics. These dynamical systems are usually called hybrid systems.
The area of hybrid systems is a fascinating discipline bridging control engineering,
computer science, and applied mathematics. A switched system is defined as a dy-
namical system with a finite number of continuous time subsystems and a logical
rule that orchestrates switching between them [33]. The need for switching usually
arises from the fact that no single candidate controller would be capable, by itself, of
guaranteeing stability and good performance when connected with a poorly modelled
process [34]. Switched system have numerous applications in control of mechani-
cal systems, process control, the automotive industry, switching power converters,
aircraft and traffic control, and many other fields [35].
Mathematically, a switched system can be described by a differential equation
of the form
x(t) = fσ(t)(x(t)) (1.3)
where x(t) ∈ Rn is the state. {fσ(t) : σ(t) ∈ P} is a family of sufficiently regular (at
18
least locally Lipschitz) functions from Rn to R
n. σ(t) is a piecewise constant function
of time: [0,∞) → P, called a switching signal, with P = {1, 2, · · · , m} being a finite
index set. The function σ(t) has a finite discontinuities (switching times/instances) on
every bounded time interval and takes a constant value on every interval between two
consecutive switching times. Normally σ(t) is assumed to be continuous from the right
everywhere: σ(t) = limτ→t+ σ(τ) for each τ ≥ 0. Functions f1(·), f2(·), · · · , fm(·)
are assumed to satisfy
f1(0) = f2(0) = · · · = fm(0) = 0 (1.4)
A switched system described by (1.3) and (1.4) imply that all its subsystems have
the same state and the same equilibrium point at the origin. If all the individual
subsystems are linear time-invariant, a switched linear system can be obtained,
x(t) = Aσ(t)x(t) (1.5)
A main concern in the switched systems design is the stability issue. There are
examples in [36] showing that unconstrained switching may destabilize a switched
system even if all individual subsystems are stable. It also may be possible to stabilize
a switched system by means of suitably constrained switching even if all individual
subsystems are unstable. Liberzon and Morse [37] formulated three basic problems
in stability and design of switched systems.
19
A. Find conditions that guarantee asymptotic stability of the switched system (1.3)
for arbitrary switching signals.
B. Identify those classes of switching signals for which the switched system (1.3)
is asymptotically stable.
C. Construct a switching signal that makes the switched system (1.3) asymptoti-
cally stable.
Instead of just asymptotic stability for each particular switching signal, a stronger
property is desirable, namely, asymptotic or exponential stability that is uniform over
the set of all switching signals. The switched system (1.3) is uniformly asymptotically
stable if there exist a positive constant δ and a class KL function β such that for all
switching signals σ(t) the solutions of (1.3) with ‖x(0)‖ ≤ δ satisfy the inequality
‖x(t)‖ ≤ β(‖x(0)‖, t), ∀t ≥ 0 (1.6)
where ‖x‖ is referred to the Euclidean norm of vector x. If the function β takes the
form β(r, s) = cre−λs for some c, λ > 0, so that
‖x(t)‖ ≤ ce−λt‖x(0)‖, ∀t ≥ 0 (1.7)
then the system (1.3) is called uniformly exponentially stable. The switched system
has stability margin λ.
20
Much of the work on Problem A has been focused on the existence of a common
Lyapunov function (CLF). It is well known that the switched system (1.3) is uniformly
asymptotically stable for arbitrary switching signals if all of its subsystems share a
radially unbounded common Lyapunov function. Particularly for the switched linear
system (1.5), its uniform exponential stability under arbitrary switching is equivalent
to the existence of a continuously differentiable common Lyapunov function V (x) for
its constituent systems [38].
Most of the available results for the arbitrary switching problem are related
to the existence of common quadratic Lyapunov functions (CQLF). The function
V (x) = xTPx is a CQLF for the switched linear system (1.5) if (i) P is a positive def-
inite symmetric matrix, and (ii) ATp P +PAp < 0, (p ∈ P). These two conditions are
equivalent to a system of linear matrix inequalities (LMIs) in P . Thus, determining
whether or not the system (1.5) possess a CQLF amounts to checking the feasibility
of a system of LMIs. Solvers for LMIs are built on convex optimization algorithms
that quickly converge [39]. However, LMI-based methods are not effective when the
number of constituent systems is very large and they cannot be directly applied to
check the existence of a CQLF for an infinite family of systems. An alternative nu-
merical technique based on iterative gradient descent methods is presented in [40],
which can be combined with randomization algorithms and applied to compact, pos-
sibly infinite, family of system matrices Ap. In this case, it has been shown that
the algorithm will converge to a CQLF with probability one, provided such a CQLF
21
exists. For some families of linear systems, certain algebraic properties have proven
to be sufficient for the existence of CQLFs. In [37], it is shown that if all matrices in
the set {Ap : p ∈ P} commute or have a solvable Lie algebra, a CQLF exists and the
switched linear system is uniformly exponentially stable under arbitrary switching. A
brief survey on various attempts to derive the algebraic conditions for the existence
of a CQLF is given in [41].
In general, CQLF existence is only a sufficient condition for the exponential
stability of a switched linear system under arbitrary switching. And the common
Lyapunov function may not necessary be quadratic. Some of the research [42, 43, 44,
45] has focused on the existence conditions and construction of common piecewise
linear Lyapunov functions (PLLFs) (also known as polyhedral Lyapunov functions).
A PLLF is of the form
V (x) = maxp∈P
(wTp x) (1.8)
where wp ∈ Rn, p ∈ P and the linear functions wT
p x are called generators of the
PLLF. However, the computational requirements to establish the common PLLF
existence is a serious bottleneck in practice. The main reason is that a complex
representation (with a large number of parameters) is usually required for a solution
to be found rendering the techniques applicable to low-dimensional problems only
[38].
One necessary condition for the switched system (1.3) being uniformly asymp-
22
totically stable under arbitrary switching is that all constituent systems are asymp-
totically stable. There are some switched systems that become unstable under certain
switching signals, even if all their constituent systems are asymptotically stable. In
order to achieve stability, one often needs to restrict the class of admissible switching
signals. This leads us to Problem B.
It is well known that a switched system is stable if all individual subsystems
are stable and the switching is sufficiently slow, so as to allow the transient effects
to dissipate after each switch. A notable approach to formulate this idea is the dwell
time approach. The concept of dwell time is introduced to specify slow switching.
Let S[τd] denote the set of all admissible switching signals with interval between
consecutive switching instances no smaller than a positive constant τd. This constant
τd is called the dwell time. Morse [46] showed that if the switching signal σ(t) of
switched linear system (1.5) “dwells” at each of its values in P long enough for the
norm of the state transition matrix of Ap, p ∈ P to drop to one in value (i.e., at least
τd time units), then Aσ will be exponentially stable with a decay rate no longer than
the smallest of the decay rates of Ap, p ∈ P. If for each p ∈ P, Ap is stable, and
‖eApt‖ ≤ e(ap−λpt), t ≥ 0, (1.9)
23
where ap ≥ 0 and λp > 0 are two finite numbers, a lower bound on τd is given by
τd > supp∈P
(apλp
)(1.10)
Dwell time switching requires each constituent system to be active for, at least,
τd units of time. Specifying a dwell time may be too restrictive for some systems. It
is possible that some subsystems may lead to bad transient responses or unacceptable
performances during such time interval. An even worse scenario is finite escape may
occur for some nonlinear subsystems before the next switching is permitted. Thus it
is of interest to relax the concept of dwell time, allowing the possibility of switching
fast when necessary and then compensating for it by switching sufficiently slowly
later. The concept of average dwell time from [34] serves this purpose. The switching
signal set is enlarged by including signals with switching intervals occasionally smaller
than a positive constant τd. But the average of all intervals is no less than τd. For
each switching signal σ and each t ≥ τ ≥ 0, let Nσ(t, τ) denote the number of
discontinuities of σ in the open interval (τ, t). For two given positive numbers τd and
N0, denote S[τd, N0] to be the set of all switching signals for which
Nσ(t, τ) ≤ N0 +t− τ
τd(1.11)
The constant τd is called the average dwell time and N0 the chatter bound.
Consider the switched linear system (1.5), given a positive constant λ0 such that
24
(Ap + λ0I) is asymptotically stable for each p ∈ P, Hespanha and Morse [34] proved
that for any λ ∈ [0, λ0), there is a finite constant τ∗d such that system (1.5) is uniformly
exponentially stable over S[τd, N0] with stability margin λ, for any average dwell time
τd ≥ τ∗d and any chatter boundN0 > 0. Since each (Ap+λ0I) is asymptotically stable,
there exists a set of symmetric, positive definite matrices {Qp : p ∈ P}, such that
Qp(Ap + λ0I) + (Ap + λ0I)TQp = −I, p ∈ P (1.12)
Let
µ = supi,j∈P
σmax[Qi]
σmin[Qj ](1.13)
where σmax[Q] and σmin[Q] denote the largest and smallest singular values of Q
respectively. After choosing a stability margin λ, one will obtain a lower bound of
the average dwell time as
τ∗d =lnµ
2(λ0 − λ)(1.14)
Similar results for certain classes of switched nonlinear systems were also derived in
[34].
The average dwell time approach discussed above mainly focuses on switched
linear systems consisting of only Hurwitz stable subsystems. However, in practice,
25
there are cases where switching to unstable subsystems becomes unavoidable. There-
fore, slow switching is not sufficient for stability. It is also required that the switched
system does not spend too much time in the unstable subsystems. In [47], Hu et
al. showed that if both Hurwitz stable and unstable subsystems exist, the switched
system (1.5) is exponentially stable under the assumption that all subsystem ma-
trices are pairwise commutative. Zhai et al. [48] presented an average dwell time
based method to analyze the stability of switched systems consisting of both Hurwitz
stable and unstable subsystems. If the total activation time ratio between Hurwitz
stable subsystems and unstable subsystems is no less than a specified constant, then
exponential stability of a desired margin is guaranteed. By introducing the concepts
of positive stability margin and negative stability margin for Hurwitz stable and un-
stable subsystems respectively, a lower bound of the activation time ratio can be
computed. Then the average dwell time results from [34] can be applied to prove the
exponential stability.
We can see that in both [34] and [48], a series of Lyapunov equations are solved
for deriving the lower bound of the average dwell time. The solutions of these Lya-
punov equations can be used to construct quadratic Lyapunov functions. Such Lya-
punov functions are closely related to the stability of the switched systems. This leads
us to another approach for stability analysis: multiple Lyapunov functions (MLF) ap-
proach. Rather than using a single common Lyapunov function that applies to all
subsystems at all time, MLF approach requires that each subsystem has its corre-
26
sponding Lyapunov or Lyapunov-like function combined with a bounding condition
at each switching time. These multiple functions can be pieced together in some way
to produce a non-traditional global Lyapunov function whose overall energy decreases
to zero along the system state trajectories [49].
Among all the published work about this approach, Branicky presented a very
intuitive version in [50]. Branicky’s basic idea was to first define a family of Lyapunov-
like functions {Vp : p ∈ P} corresponding to all subsystems of system (1.3). Instead
of being defined globally, each Vp(x) is defined over the region where subsystem p is
active. Each function Vp(x) also needs to satisfy two conditions: (i) positive definite,
i.e., Vp(x) > 0, ∀x 6= 0, and Vp(0) = 0; (ii) non-positive definite time derivative,
Vp(x) ≤ 0. If the values of Vp(x) at the entry points (when subsystem p is switched
on) are monotonically nonincreasing for all p ∈ P, the equilibrium point x = 0 of
system (1.3) is stable in the sense of Lyapunov. Same result can be obtained if Vp(x)
is monotonically nonincreasing at the end points. Zhai et al. [51] complemented this
result by introducing the idea of evaluating “the average value of multiple Lyapunov
functions”. For subsystem p, during each of its active time interval, an average value
of Vp(x) over this active time interval is calculated. A sequence of these average values
corresponding to subsystem p is then used to evaluate the system’s stability.
The requirements of Lyapunov-like function in Branicky’s result is weakened in
[52]. Instead of having non-positive definite derivative, Vp(x) only has to be bounded
by a continuous function which is zero at the origin. Vp(x) is called a weak Lyapunov-
27
like function if it is monotonically nonincreasing at the entry points (or end points).
The equilibrium point x = 0 of system (1.3) is stable in the sense of Lyapunov if such
weak Lyapunov-like functions exist for each subsystem.
A more general stability result using MLF approach is proposed in [53]. It is
proved that asymptotic stability is guaranteed by requiring the sequence of values of
the Lyapunov-like function candidate at consecutive switching times to be decreasing
and the Lyapunov-like function between these times is bounded by a continuous
function which is zero at the origin.
Using MLFs to form a single non-traditional Lyapunov function offers much
greater freedom and infinitely more possibilities for demonstrating stability, for con-
structing a non-traditional Lyapunov function, and for achieving the stabilisation
of the switched system [49]. But a critical challenge of applying MLF theory in
practical switched systems is how to construct a proper family of Lyapunov-like
functions. Currently, there is no universal constructive procedure for choosing the
best Lyapunov-like functions. However, if one focuses on the linear cases, piecewise
quadratic Lyapunov-like function could be an attractive candidate, since the stability
conditions can be formulated as LMIs. In [54], the search for continuous piecewise
quadratic Lyapunov-like functions is formulated as a convex optimization problem in
terms of LMIs.
All the above discussed MLF approaches have a common restriction in which
all the candidate Lyapunov-like functions must satisfy the same set of conditions.
28
This will limit the flexibility of choosing proper candidate Lyapunov-like functions
for each subsystem accordingly.
The dwell time and MLF techniques not only can be applied for stability anal-
ysis, also give the means of constructing stabilizable switching signals, which provide
solutions to Problem C given earlier.
1.3 Introduction of Algorithms
The work presented in this thesis is mainly based on two control algorithms which
are briefly introduced here.
1.3.1 Adaptive internal model principle control algorithm
Brown and Zhang developed an IMP based adaptive algorithm to identify unknown
frequencies of periodic disturbances in [55, 56]. In this algorithm, a standard IMP
controller is implemented using a state space representation as follows
x1
x2
=
0 ω
−ω 0
x1
x2
+
0
Kf
e(t) (1.15)
u =
[0 1
]x1
x2
(1.16)
29
where x1, x2 are the two state variables of the controller, the tracking error e(t) is the
input, and u is the output control signal. Kf is a tuning gain of the controller. Since
the signal frequency ωc is unknown, according to the certainty equivalence principle,
its best estimate ω is used in (1.15). A mapping from the controller’s state variables
to the frequency estimation error ε can be derived as
ε = −Kfex1
x21 + x22(1.17)
Ideally, this estimation error can be eliminated in one updating step if the exogenous
input is noise-free pure sinusoid. However, this is not a realistic situation. An integral
controller was used for eliminating the error as follows,
dω
dt= −Kω
Kfex1
x21 + x22(1.18)
where Kω is the integral gain. Thus the parameter ω in the IMP controller is con-
stantly updated and asymptotically converges to the true frequency ωc.
As an alternative approach, a least-squares method was also introduced in [55]
to estimate the frequency ωc, which is given by
dω
dt= − a1
(a2t+ 1)×
Kfex1
x21 + x22(1.19)
where a1 is chosen as the estimate of the uncertainty in initial knowledge of ωc, and
30
a2 = a1ψ with ψ being the variance of (ε/Kf ). This approach is only appropriate
when the disturbance frequency does not drift with time. By using a Kalman filter,
the adaptation gain in (1.19) is replaced bya1+a3ta2t+1 , so that the estimated frequency
can be time-varying. With the identification of its frequency, the periodic disturbance
is perfectly rejected and zero-error regulation is achieved. The asymptotic stability of
this adaptive algorithm was proved using singular perturbation theory and averaging
theory.
By placing multiple number of this adaptive IMP controllers in parallel in the
feedback path, a periodic signal consisting of multiple sinusoids with different un-
known frequencies can be identified and cancelled. According to the IMP, each IMP
controller corresponds to a sinusoidal component of the signal. One such application
is the power system frequency estimation. In [56], five adaptive IMP controllers are
constructed to cancel a signal with 1st to 9th odd harmonics. All five controllers share
the same estimated fundamental frequency. In [55], two adaptive IMP controllers are
implemented to identify the frequencies of a signal with two non-harmonic compo-
nents. Compared to the results presented in [57], which uses an adaptive observer
technique to identify multiple frequencies, this adaptive IMP algorithm has much
faster convergence.
31
1.3.2 Intermittent cancellation control algorithm
When a system is subjected to both a narrowband disturbance and an additive white
noise, a traditional IMP controller and a wideband disturbance controller can be
used together. Since the IMP controller introduces a phase lag into the system, it
will limit the wideband disturbance controller’s capability of minimizing white noise.
Brown et al. proposed an intermittent cancellation control algorithm in [2, 3]. A
wideband disturbance controller is always active in this control algorithm. An IMP
controller is active for narrowband disturbance rejection only when the tracking error
is significant. The input to the IMP controller is disconnected when the tracking
error is smaller than a predefined threshold as a result of near perfect rejection of
narrowband disturbance. Note that the IMP controller continues to provide an out-
put, which after convergence provides perfect cancellation of the predictable portion
of the disturbance or necessary control action to track the reference. Residual errors
will cause the switching mechanism to infrequently close the switch improving the
performance of the system with each invocation. It can be seen that when the switch
is closed, we have a closed-loop IMP control system, and when the switch is open, the
IMP controller provides an open loop cancellation of the error. Thus this intermit-
tent cancellation control system can be modelled as a switched system which switches
between open loop and closed-loop control. With this switching mechanism, the wide-
band disturbance controller can be made more aggressive for attenuating white noise
while maintaining the stability margins and control actions. When the narrowband
32
disturbance is constant, an integral controller is the IMP controller. A traditional
IMP controller is implemented when the narrowband disturbance is a known sinu-
soid. If the narrowband disturbance is a sum of known number of sinusoids, multiple
IMP controllers can be placed in parallel for the rejection.
1.4 Contributions of the Thesis
Most of the internal model principle based approaches in regulation problems focus
on periodic signals. In this thesis, two advanced applications of the internal model
principle control theory are developed.
First, the IMP based adaptive algorithm in [55] is extended to identify EDS
signals with unknown parameters in discrete time. Since the amplitude of an EDS
signal varies exponentially, an additional parameter, damping factor, needs to be
estimated as well as the frequency. In this algorithm, a discrete time state space model
of an EDS signal is derived based on its continuous time version. Estimated signal
parameters are used in this model according to the certainty equivalence principle.
After two mappings between the parameter estimation errors and the states of the
state space model are derived, two integral controllers are implemented to update the
estimated parameters respectively. This control law results in a system output error
that decays exponentially fast with a decay rate independent of the signal’s damping
factor. A discrete time two time scale averaging theory developed in [58] is used to
prove the stability and convergence of the adaptive algorithm. Simulation results
33
show that this algorithm is capable of tracking time-varying parameters or a signal
with multiple EDS components. This work is presented in Chapter 2, and has been
published in the Proceedings of the 17th IFAC World Congress [59].
The stability analysis of the intermittent control algorithm presented in [2] was
left as an open issue, which is addressed in Chapter 3 in this thesis. The intermittent
control system is first modelled as a switched system with an open loop control
subsystem and a closed-loop control subsystem. In order to carry out the stability
analysis of the switched system model, an extended multiple Lyapunov functions
approach is developed to relax some constraints imposed by existing results. In this
new approach, not all constituent systems are required to have stable equilibria, as
long as those subsystems satisfy a bounding condition. This allows the subsystems to
not share a common equilibrium point. Thus subsystems will be defined as not stable
if they have unstable equilibrium points or lack the necessary equilibrium point. The
not stable subsystems are also allowed to be active for arbitrary long periods of time.
In addition, this extension allows the system to switch between controllers having
different dimensions or states. The extended MLF theorem and stability analysis of
the intermittent integral control system are presented in Chapter 3, and have been
published in the Proceedings of the 2010 American Control Conference [60, 61]. In
addition, some simulation results are also presented in Chapter 3, which are included
in a submission to the IEEE Transactions on Automatic Control [62].
The second advanced implementation of the internal model principle is to com-
34
bine the intermittent control with the adaptive IMP control algorithm introduced
in Section 1.3. This extended switching control strategy can be applied to reject a
periodic disturbance with unknown frequencies. The periodic disturbance can be can-
celled once its frequencies are identified by the adaptive IMP control algorithm. Thus
the input to the adaptive IMP controller is disconnected, and may be reconnected in-
termittently due to the residue error. A switched system model is formulated as well
as a stability theorem. This extended intermittent cancellation control is presented
in Chapter 4. Part of this chapter is submitted to the 24th Canadian Conference on
Electrical and Computer Engineering [63].
1.5 Organization of the Thesis
In Chapter 2, a discrete time version internal model principle based adaptive al-
gorithm is developed for the identification of exponentially damped sinusoids with
unknown parameters. This algorithm has two updating laws, one for the estimated
frequency, and one for the estimated damping factor. By using a two time scale av-
eraging theory, this adaptive algorithm is proved to be locally exponentially stable.
Three MATLAB/Simulink simulation examples are also presented to illustrate the
performance of the algorithm.
In Chapter 3, a combination of open and closed-loop control strategy is pre-
sented for attenuating disturbances with both constant component and white Gaus-
sian noise. The system switches between a state feedback control and an augmented
35
state feedback control with integral action. A multiple Lyapunov functions theorem
is developed as an extension to existing approaches. This theorem is then used to
prove the stability of the introduced system which is modelled as a switched system.
A numerical example is presented to demonstrate the performance of this switching
control strategy, as well as comparison with two other traditional controllers.
This switching control strategy is extended in Chapter 4, where the closed-loop
mode controller includes a state feedback controller and an adaptive IMP controller.
Thus this control strategy can be applied to minimize disturbances consist of white
noise and a periodic disturbance with unknown frequencies. A switched system model
is derived as well as a stability theorem. Simulation results are presented to demon-
strate the performance improvement. In Chapter 5, some concluding remarks are
drawn.
Bibliography
[1] B. A. Francis and W. M. Wonham, “The Internal Model Principle of Control
Theory,” Automatica, vol. 12, pp. 457–465, 1976.
[2] L. J. Brown, G. E. Gonye, and J. S. Schwaber, “Non-linear PI Control Inspired
by Biological Control Systems,” in Proceedings of the 37th IEEE Conference on
Decision and Control, Tampa, FL, Dec. 1998, pp. 1040–1045.
[3] L. J. Brown and J. S. Schwaber, “Intermittent Cancellation Control: A Control
Paradigm Inspired by Mammalian Blood Pressure Control,” in Proceedings of
the American Control Conference, San Diego, CA, Jun. 1999, pp. 139–143.
[4] N. E. Huang and Z. Shen, et al., “The Empirical Mode Decomposition and the
Hilbert Spectrum for Nonlinear and Non-Stationary Time Series Analysis,” in
Procedures of the Royal Society of London, vol. A454, 1998, pp. 903–995.
[5] L. Cohen, Time-Frequency Analysis, 1st ed. Englewood Cliffs, NJ: Prentice
Hall, 1995.
[6] R. L. Streit and R. F. Barrett, “Frequency Tracking Using Hidden Markov Mod-
els,” IEEE Transactions on Acoustics, Speech and Signal Processing, vol. 38, pp.
586–598, Apr. 1990.
[7] L. B. White, “Cartesian Hidden Markov Models with Applications,” IEEE
Transactions on Signal Processing, vol. 40, pp. 1601–1604, Jun. 1992.
[8] H. K. Kwok and D. L. Jones, “Improved Instantaneous Frequency Estimation
Using an Adaptive Short-Time Fourier Transform,” IEEE Transactions on Signal
Processing, vol. 48, no. 10, pp. 2964–2972, Oct. 2000.
[9] B. F. La Scala and R. R. Bitmead, “Design of an Extended Kalman Filter Fre-
quency Tracker,” IEEE Transactions on Signal Processing, vol. 44, no. 3, pp.
739–742, Mar. 1996.
36
37
[10] S. Bittanti and S. Savaresi, “On the Parameterization and Design of an Extended
Kalman Filter Frequency Tracker,” IEEE Transactions on Automatic Control,
vol. 45, no. 9, pp. 1718–1724, Sep. 2000.
[11] G. Li, “A Stable and Efficient Adaptive Notch Filter for Direct Frequency Esti-
mation,” IEEE Transactions on Signal Processing, vol. 45, no. 8, pp. 2001–2009,
Aug. 1997.
[12] P. A. Regalia, “An Improved Lattice-Based Adaptive IIR Notch Filter,” IEEE
Transactions on Signal Processing, vol. 39, no. 9, pp. 2124–2128, Sep. 1991.
[13] M. V. Dragosevic and S. S. Stankovic, “An Adaptive Notch Filter with Improved
Tracking Properties,” IEEE Transactions on Signal Processing, vol. 43, no. 9, pp.
2068–2078, Sep. 1995.
[14] M. Bodson and S. C. Douglas, “Adaptive Algorithms for the Rejection of Sinu-
soidal Disturbances with Unknown Frequency,” Automatica, vol. 33, no. 12, pp.
2213–2221, 1997.
[15] B. Wu and M. Bodson, “A Magnitude/Phase-Locked Loop Approach to Param-
eter Estimation of Periodic Signals,” IEEE Transactions on Automatic Control,
vol. 48, no. 4, pp. 612–618, Apr. 2003.
[16] R. Roy and T. Kailath, “ESPRIT-Estimation of Signal Parameters via Rota-
tional Invariance Techniques,” IEEE Transactions on Acoustics, Speech, and
Signal Processing, vol. 37, no. 7, pp. 984–995, Jul. 1989.
[17] R. Roy, A. Paulraj, and T. Kailath, “Comparative Performance of ESPRIT and
MUSIC for Direction-Of-Arrival Estimation,” in IEEE International Conference
on Acoustics, Speech and Signal Processing, Apr. 1987, pp. 2344–2347.
[18] T. C. Tsao, Y. X. Qian, and M. Nemani, “Repetitive Control for Asymptotic
Tracking of Periodic Signals with an Unknown Period,” Journal of Dynamic
Systems, Measurement, and Control, vol. 122, pp. 364–369, Jun. 2000.
[19] A. Serrani, A. Isidori, and L. Marconi, “Semiglobal Nonlinear Output Regula-
tion with Adaptive Internal Model,” IEEE Transactions on Automatic Control,
vol. 46, no. 8, pp. 1178–1194, Aug. 2001.
38
[20] Z. Ding, “Global Stabilization and Disturbance Suppression of a Class of Non-
linear System with Uncertain Internal Model,” Automatica, vol. 39, pp. 471–479,
2003.
[21] R. Marino and P. Tomei, “Output Regulation for Linear Systems via Adaptive
Internal Model,” IEEE Transactions on Automatic Control, vol. 48, no. 12, pp.
2199–2202, Dec. 2003.
[22] I. D. Landau, A. Constantinescu, and D. Rey, “Adaptive narrow band distur-
bance rejection applied to an active suspension - an internal model approach,”
Automatica, vol. 41, pp. 563–574, 2005.
[23] R. Boyer, S. Essid, and N. Moreau, “Non-Stationary Signal Parametric Modeling
Techniques with an Application to Low Bitrate Audio Coding,” in Proceedings
of the 6th International Conference on Signal Processing, Aug.26-30, 2002, pp.
430–433.
[24] R. Boyer and J. Rosier, “Iterative Method for Harmonic and Exponentially
Damped Sinusoidal Models,” in Proceedings of the 5th International Conference
on Digital Audio Effects, Hamburg, Germany, Sep.26-28, 2002, pp. 145–150.
[25] Y. Hua and T. K. Sarkar, “Matrix Pencil Method for Estimating Parameters of
Exponentially Damped/Undamped Sinusoids in Noise,” IEEE Transactions on
Acoustics, Speech, and Signal Processing, vol. 38, no. 5, pp. 814–824, May 1990.
[26] M. Jeanneau, P. Mouyon, and C. Pendaries, “Estimating the Parameters of Ex-
ponentially Damped Sinusoids in Light and Flexible Structures using the SIN-
TRACKMethod,” in Proceedings of the 13th Southeastern Symposium on System
Theory, Morgantown, WV, Mar.8-10 1998, pp. 115–119.
[27] R. Badeau, R. Boyer, and B. David, “EDS Parametric Modeling and Tracking
of Audio Signals,” in Proceedings of the 5th International Conference on Digital
Audio Effects, Hamburg, Germany, Sep.26-28, 2002, pp. 1–6.
[28] A. Gunnarsson and I. Y. Gu, “Music Signal Synthesis Using Sinusoid Models
and Sliding-Window ESPRIT,” in Proceedings of the 2006 IEEE International
Conference on Multimedia and Expo, Toronto, ON, Jul.9-12, 2006, pp. 1389–
1392.
39
[29] Y. Bresler and A. Macovski, “Exact Maximum Likelihood Parameter Estimation
of Superimposed Exponential Signals in Noise,” IEEE Transactions on Acoustics,
Speech, and Signal Processing, vol. ASSP-34, no. 5, pp. 1081–1089, Oct. 1986.
[30] C. K. Papadopoulos and C. L. Nikias, “Parameter Estimation of Exponentially
Damped Sinusoids Using Higher Order Statistics,” IEEE Transactions on Acous-
tics, Speech, and Signal Processing, vol. 38, no. 8, pp. 1424–1436, Aug. 1990.
[31] R. Kumaresan and D. W. Tufts, “Estimating the Parameters of Exponentially
Damped Sinusoids and Pole-zero Modeling in Noise,” IEEE Transactions on
Acoustics, Speech, and Signal Processing, vol. ASSP-30, no. 6, pp. 833–840, Dec.
1982.
[32] J. Zhang, A. K. Swain, and S. K. Nguang, “Parameter Estimation of Exponen-
tially Damped Sinusoids using HSVD based Extended Complex Kalman Filter,”
in Proceedings of 2008 IEEE Region 10 Conference - TENCON 2008, Hyderabad,
India, Nov.18-21 2008, pp. 1–6.
[33] A. V. Savkin and R. J. Evans, Hybrid Dynamical Systems: Controller and Sensor
Switching Problems. Boston, MA: Birkhauser, 2002.
[34] J. P. Hespanha and A. S. Morse, “Stability of Switched Systems with Aver-
age Dwell-time,” in Proceedings of the 38th IEEE Conference on Decision and
Control, Phoenix, AZ, Dec. 1999, pp. 2655–2660.
[35] A. S. Morse, Ed., Control Using Logic-Based Switching, 1st ed. London, UK:
Springer-Verlag, 1997.
[36] D. Liberzon, Switching in Systems and Control. Boston, MA: Birkhauser, 2003.
[37] D. Liberzon and A. S. Morse, “Basic Problems in Stability and Design of
Switched Systems,” IEEE Control Systems Magazine, vol. 19, pp. 59–70, 1999.
[38] R. Shorten, F. Wirth, O. Mason, and K. Wulff, “Stability Criteria for Switched
and Hybrid Systems,” SIAM Review, vol. 49, no. 4, pp. 545–592, Nov. 2007.
[39] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities
in System and Control Theory. Philadelphia, PA: SIAM, 1994.
40
[40] D. Liberzon and R. Tempo, “Common Lyapunov Functions and Gradient Algo-
rithms,” IEEE Transactions on Automatic Control, vol. 49, no. 6, pp. 990–994,
2004.
[41] H. Lin and P. J. Antsaklis, “Stability and Stabilizability of Switched Linear Sys-
tems: A Survey of Recent Results,” IEEE Transactions on Automatic Control,
vol. 54, no. 2, pp. 308–322, Feb. 2009.
[42] R. K. Brayton and C. H. Tong, “Constructive Stability and Asymptotic Stabil-
ity of Dynamical Systems: A Constructive Approach,” IEEE Transactions on
Circuits and Systems, vol. 27, no. 11, pp. 1121–1130, Nov. 1980.
[43] H. Kiendl, J. Adamy, and P. Stelzner, “Vector Norms as Lyapunov Functions for
Linear Systems,” IEEE Transactions on Automatic Control, vol. 37, no. 6, pp.
839–842, Jun. 1992.
[44] F. Blanchini, “Nonquadratic Lyapunov Functions for Robust Control,” Auto-
matica, vol. 31, no. 3, pp. 451–461, 1995.
[45] A. Polanski, “On Absolute Stability Analysis by Polyhedral Lyapunov Func-
tions,” Automatica, vol. 36, no. 4, pp. 573–578, Apr. 2000.
[46] A. S. Morse, “Supervisory Control of Families of Linear Set-Point Controllers–
Part 1: Exact Matching,” IEEE Transactions on Automatic Control, vol. 41,
no. 10, pp. 1413–1431, Oct. 1996.
[47] B. Hu, X. Xu, A. N. Michel, and P. J. Antsaklis, “Stability Analysis for a Class
of Nonlinear Switched Systems,” in Proceedings of the 38th IEEE Conference on
Decision and Control, Phoenix, AZ, Dec. 1999, pp. 4374–4379.
[48] G. Zhai, B. Hu, K. Yasuda, and A. N. Michel, “Stability Analysis of Switched
Systems with Stable and Unstable Subsystems: An Average Dwell Time Ap-
proach,” in Proceedings of the American Control Conference, Chicago, IL, Jun.
2000, pp. 200–204.
[49] R. A. DeCarlo, M. S. Branicky, S. Pettersson, and B. Lennartson, “Perspectives
and Results on the Stability and Stabilizability of Hybrid Systems,” Proceedings
of the IEEE, vol. 88, no. 7, pp. 1069–1082, Jul. 2000.
41
[50] M. S. Branicky, “Multiple Lyapunov Functions and Other Analysis Tools for
Switched and Hybrid Systems,” IEEE Transactions on Automatic Control,
vol. 43, no. 4, pp. 475–482, Apr. 1998.
[51] G. Zhai, I. Matsune, J. Imae, and T. Kobayashi, “A Note on Multiple Lya-
punov Functions and Stability Condition for Switched and Hybrid Systems,” in
Proceedings of the 16th IEEE International Conference on Control Applications,
Singapore, Oct.1-3, 2007, pp. 226–231.
[52] H. Ye, A. N. Michel, and L. Hou, “Stability Analysis of Discontinuous Dynamical
Systems with Applications,” in Proceedings of the 13th World Congress of the
International Federation of Automatic Control, vol. E, San Francisco, CA, Jun.
1996, pp. 461–466.
[53] ——, “Stability Theory for Hybrid Dynamical Systems,” IEEE Transactions on
Automatic Control, vol. 43, no. 4, pp. 461–474, Apr. 1998.
[54] M. Johansson and A. Rantzer, “Computation of Piecewise Quadratic Lya-
punov Functions for Hybrid Systems,” IEEE Transactions on Automatic Control,
vol. 43, no. 4, pp. 555–559, Apr. 1998.
[55] L. J. Brown and Q. Zhang, “Identification of Periodic Signals with Uncertain
Frequency,” IEEE Transactions on Signal Processing, vol. 51, no. 6, pp. 1538–
1545, Jun. 2003.
[56] L. J. Brown and Q. Zhang, “Periodic Disturbance Cancellation with Uncertain
Frequency,” Automatica, vol. 40, pp. 631–637, 2004.
[57] R. Marino and P. Tomei, “Global Estimation of n Unknown Frequencies,” IEEE
Transactions on Automatic Control, vol. 47, no. 8, pp. 1324–1328, Aug. 2002.
[58] E. Bai, L. Fu, and S. S. Sastry, “Averaging Analysis for Discrete Time and
Sampled Data Adaptive Systems,” IEEE Transactions on Circuits and Systems,
vol. 35, no. 2, pp. 137–148, Feb. 1988.
[59] J. Lu and L. J. Brown, “Identification of Exponentially Damped Sinusoidal Sig-
nals,” in Proceedings of the 17th IFAC World Congress, Seoul, South Korea,
Jul.6-11, 2008, pp. 5089–5094.
42
[60] ——, “A Multiple Lyapunov Functions Approach for Stability of Switched Sys-
tems,” in Proceedings of the 2010 American Control Conference, Baltimore, MD,
Jul. 2010, pp. 3253–3256.
[61] ——, “Stability Analysis of a Proportional with Intermittent Integral Control
System,” in Proceedings of the 2010 American Control Conference, Baltimore,
MD, Jul. 2010, pp. 3257–3262.
[62] ——, “Combining Open and Closed-Loop Control via Intermittent Integral Con-
trol Action,” IEEE Transactions on Automatic Control, submitted.
[63] ——, “A Combination of Open and Closed-Loop Control for Disturbance Rejec-
tion,” The 24th Canadian Conference on Electrical and Computer Engineering,
submitted.
43
Chapter 2
Identification of Exponentially Damped
Sinusoidal Signals 1
2.1 Introduction
In this chapter, we consider signals composed of a sum of exponentially damped
sinusoids with the following form,
s(k) =N∑
i=1
si(k) =N∑
i=1
aie−σik sin(ωik + ϕi) (2.1)
where the uncertain σi and ωi are the damping factor and the frequency, respectively.
This form can also represent constant-amplitude sinusoids, and constant signals. The
objective is to estimate the parameters σi and ωi. There have been many techniques
developed to deal with predictable signals, such as narrowband or sinusoidal signals,
1. This chapter has been published.J. Lu and L. J. Brown, “ Identification of Exponentially Damped Sinusoidal Signals”, Proc.of the 17th IFAC World Congress, Seoul, July 6-11, 2008, pp. 5089-5094.
44
since they appear in both signal processing and control applications, including ac-
tive noise control, radar signals, rotating mechanical systems, computer hard disk
drive etc. [1]. These techniques include linear quadratic regulator based modern
control, higher harmonic control [2], adaptive notch filter [3], adaptive feedforward
cancellation (AFC) [4], adaptive observer technique [5].
Another common approach for perfect cancellation of signals is based on a
fundamental control principle, the internal model principle (IMP) [6]. This principle
states that perfect disturbance rejection or reference tracking is achieved when a
model of the dynamic structure of the disturbance or reference signal is incorporated
in a stable feedback loop. The accuracy of regulation depends critically on the fidelity
of the IMP controller. Errors of less than one percent in model coefficients can
lead to unacceptable residual errors. Thus, the ability to adaptively tune the model
parameters, which can be completely specified as damping factors and frequencies, is
of great benefit. Then adaptive IMP controllers can provide exact reproduction of the
predictable part of a signal, and when they do, they provide highly accurate estimates
of the signal parameters. One application of the IMP to periodic disturbance rejection
is repetitive control for time-lag systems and multi-link manipulators [7]. Serrani et
al. [8] also presented a solution to a nonlinear output regulation problem based on
the IMP.
An IMP based adaptive algorithm for canceling quasi-periodic, or narrowband
signals with uncertain frequencies is presented in [1]. This approach begins with a
45
state space implementation of the standard IMP controller with the best estimate of
the frequency used. A simple mapping from the states of the controller to the error in
the frequency estimate was developed and this “measurement” of the frequency error
is used to update the parameters of the IMP controller. When this adaptive IMP
controller is placed into a feedback loop, the resulting closed-loop system achieves
perfect frequency estimation of the elements of a sum of sinusoidal signals.
In addition to sinusoidal signals, EDS signals are often used to model audio sig-
nals, such as speech or music, which contain relatively fast variations in amplitude.
The conventional sinusoidal model is thus extended by allowing the amplitude to
evolve exponentially as given by (2.1). A well-known approach to EDS signal param-
eter estimation is the polynomial or linear prediction method as in [9]. Traditionally,
EDS signal model is associated with a high resolution parameter estimation method,
such as matrix pencil, ESPRIT or Kung’s algorithm, [10]. Hua et al. [11] presented
a matrix pencil method as an alternative approach which exploits the structure of a
matrix pencil of the EDS signal si(t), instead of the structure of prediction equations
satisfied by si(t). In [12], the EDS signal parameters, damping factor and frequency,
are estimated using a subspace based matrix pencil high resolution method. The
tracking of the slow variation of the signal parameters is achieved using an adaptive
least mean square algorithm.
Motivated by the IMP based adaptive algorithm in [1], an extended adaptive
algorithm for EDS disturbance cancellation was developed in [13]. The control law
46
results in a system output error that decays exponentially fast with a decay rate
independent of σi. This is equivalent to what is meant when integral control is said
to provide perfect set-point tracking. This work was developed only in continuous
time framework and strictly as a control algorithm. Here we develop a discrete time
implementation of this algorithm, and convert the control algorithm into a signal
processing algorithm.
This chapter is organized as follows: In Section 2.2, the motivation, the con-
tinuous time state space representation of the IMP controller, is introduced. Then
the derivation of the IMP based adaptive algorithm for EDS signal identification in
discrete time is presented. The convergence and stability property of the proposed
adaptive algorithm is analyzed based on a discrete time two time scale averaging
theory in Section 2.3. Simulation results are demonstrated in Section 2.4, followed
by some conclusions in Section 2.5.
2.2 Internal Model Principle Based Adaptive
Algorithm
2.2.1 Adaptive algorithm in continuous time
The basic structure of the feedback system is shown in Fig. 2.1, where L is a tuning
function that is properly designed to stabilize the system, H represents the IMP
controller. d is an EDS signal, which is defined as follows,
47
+
_uh
euL H
σ, ωd
Figure 2.1: Block diagram of an internal model principle control system
d(t) = ae−σct sin(ωct+ ϕ), σc > 0, ωc > 0, a > 0 (2.2)
where σc, ωc are the damping factor and frequency, respectively. Thus, following IMP
theory, a continuous time state space representation of an IMP controller is
x1
x2
=
−σc ωc
−ωc −σc
x1
x2
+
0
1
e (2.3)
uh =
[K1 K2
]x1
x2
(2.4)
where K1, K2 are tuning gains.
If the initial conditions for x1 and x2 are given by x1(0) =a cosϕ√K21+K2
2
, x2(0) =
a sinϕ√K21+K2
2
, then for all t > 0, e = 0 and
x1(t) =ae−σct
√K21 +K2
2
cos(ωct+ ϕ) (2.5)
x2(t) =ae−σct
√K21 +K2
2
sin(ωct+ ϕ) (2.6)
48
By letting x = |x|∠θ = x1(t) + jx2(t), we can get
|x| = ae−σct√K21 +K2
2
(2.7)
θ = tan−1(x2(t)
x1(t)
)= ωct+ ϕ (2.8)
where tan−1(·) is defined to have a range given by real numbers such that θ is con-
tinuous in t. Differentiating both sides of (2.7) and (2.8), we have
σc = − 1
|x|d|x|dt
(2.9)
and
ωc =dθ
dt(2.10)
In practice, the model parameters in (2.3) and (2.4) are approximations, giving
σc = − 1
|x|d|x|dt
= −x1x1 + x2x2
x21 + x22= σ − ex2
x21 + x22(2.11)
and
ωc =dθ
dt=
d
dttan−1
(x2(t)
x1(t)
)=x1x2 − x1x2
x21 + x22= ω − ex1
x21 + x22(2.12)
49
Thus the error between σc and σ can be expressed as
σ = − ex2
x21 + x22(2.13)
Similarly, the error between ωc and ω is
ω = − ex1
x21 + x22(2.14)
(2.11) and (2.12) can be used to estimate the damping factor and frequency of the
EDS signal.
2.2.2 Derivation of the adaptive algorithm in discrete time
The discrete time state space equation of the IMP controller can be converted from
its continuous time counterpart (2.3) and (2.4) as
x1(k + 1)
x2(k + 1)
= e−σ
cosω sinω
− sinω cosω
x1(k)
x2(k)
+
0
1
e(k) (2.15)
uh(k) =
[K1 K2
]x1(k)
x2(k)
(2.16)
Therefore at sampling instant t = kTs, continuous time estimation errors (2.13)
50
and (2.14) are equal to the following discrete time estimation errors
σ(k) = − e(k)x2(k)
x21(k) + x22(k)(2.17)
ω(k) = − e(k)x1(k)
x21(k) + x22(k)(2.18)
and the estimates of the damping factor and frequency can be updated by using two
integral controllers
σ(k + 1) = σ(k) + εσ(k)
= σ(k)− εe(k)x2(k)
x21(k) + x22(k)(2.19)
ω(k + 1) = ω(k) + εKbω(k)
= ω(k)− εKbe(k)x1(k)
x21(k) + x22(k)(2.20)
where ε is a small adaptation gain, Kb is a constant, and both are positive.
From (2.19) and (2.20), it can be seen that if x1(k) = 0 and x2(k) = 0, the
integral update laws are undefined as both the numerators and denominators are 0.
This problem can be avoided by adding a small constant C in the denominators of
both equations, or setting ε = 0 when |x| is less than a constant.
Fig. 2.2 and Fig. 2.3 show the structure of the IMP based adaptive feedback
system, where AIM means adaptive IMP controller. The function Fk(x, e) is what
have been derived in (2.17) and (2.18). Due to the structure of the IMP controller,
51
L(z)
H(z)
_
d + u e
k kσ εσ+ k bK kω ε ω+
Fk(x,e)
σk+1 ωk+1
kσ kω
[ ]1 2
Tx x=x
uh
AIM
Figure 2.2: Block diagram of the adaptive IMP control system
e
unnth AIM
_
u11st AIM
Σ
_L(z)
d + u
•••
Figure 2.3: Block diagram of an adaptive IMP feedback control system for multi-EDSsignals
a controller can only achieve identification of a single EDS mode. If the signal d(k)
is composed of multiple EDS modes, multiple adaptive IMP controllers are placed in
parallel.
By incorporating the adaptive update laws (2.19)-(2.20), the IMP controller
parameters, σ and ω, converge to the true values σc and ωc of the EDS signal. If σc
or ωc changes with time, e is not zero and the adaptive algorithm will estimate and
track the changing parameters.
52
Note that for a sampled-data system, some knowledge of the EDS signal fre-
quency is necessary to determine the sampling frequency.
2.3 Convergence and Stability Analysis
The stability analysis of the proposed adaptive algorithm in continuous time is pre-
sented in [13]. Singular perturbation theorem [14], and averaging theorem [15], are
used for the analysis. Bai et al. adapted these two theorems in a combined discrete
time version, Theorem 2.2.4. Exponential Stability Theorem for Two-Time Scale Sys-
tem, in [16].
The feedback system is now formulated as a two time scale model. Due to
the limitation of space, we will address only the case where the signal is composed
of a single EDS. The techniques for extending the proof to the multi-EDS case are
shown in [1, 17]. The primary change in the proof is that the equilibria x10 and
x20 defined in equations (2.32) and (2.33) will have terms corresponding to each
mode, and the calculation of the averaging function will be far more complicated.
However, as simple sinusoids, these extra terms will ultimately contribute nothing to
the average, beyond possibly requiring longer averaging times. Detailed calculations
of the averaged function have also been omitted for space reasons. The state space
equations for the adaptive feedback system in Fig. 2.2 are as follows:
xp(k + 1) = Apxp(k) +Bpu(k) (2.21)
53
e(k) = Cpxp(k) (2.22)
u(k) = −K1x1(k)−K2x2(k) + d(k) (2.23)
d(k) = ae−σck sin(ωck) + c1 (2.24)
x1(k + 1) = (e−σ cosω)x1(k) + (e−σ sinω)x2(k) (2.25)
x2(k + 1) = −(e−σ sinω)x1(k) + (e−σ cosω)x2(k) + e(k) (2.26)
σ(k + 1) = σ(k)− εe(k)x2(k)
x21(k) + x22(k)(2.27)
ω(k + 1) = ω(k)− εKbe(k)x1(k)
x21(k) + x22(k)(2.28)
where (2.21) and (2.22) are a state space representation for L(z). The presence of
the bias c1 in the exogenous signal (2.24) does not affect the convergence property of
the algorithm if L(1) = 0.
Perturbation analysis proceeds by fixing σ(k) = σ, and ω(k) = ω. Under these
conditions, the transfer function of the IMP controller is
H(z) =K2z +K1e
−σ sinω −K2e−σ cosω
z2 − 2ze−σ cosω + e−2σ
:=N(z)
D(z).
The tuning function can also be expressed as a rational polynomial form as L(z) =
B(z)A(z)
. Thus the transfer function from d(k) to e(k) can be given by
Ged(z) =B(z)D(z)
A(z)D(z) +B(z)N(z)
54
= (z2 − 2ze−σ cosω + e−2σ)Q(z)
where Q(z) =B(z)
A(z)D(z)+B(z)N(z). The transfer functions from d(k) to x1(k) and
x2(k) can also be derived as
Gx1d(z) = (e−σ sinω)Q(z),
Gx2d(z) = (z − e−σ cosω)Q(z).
Now in order to formulate the two time scale model, we introduce new state
variables, σe, ωe, xpe, x1e, x2e, as follows:
σe(k) = σ(k)− σc (2.29)
ωe(k) = ω(k)− ωc (2.30)
xpe(k) = xp(k)− xp0(k) (2.31)
x1e(k) = x1(k)− a|Q|e−σck−σ(k) sinω(k) sin(ωck + ∠Q)
:= x1(k)− x10(k) (2.32)
x2e(k) = x2(k)− a|Q|e−σck−σ(k)[e−σc sinωc cos(ωck + ∠Q)
+ (e−σc cosωc − e−σ(k) cosω(k)) sin(ωck + ∠Q)]
:= x2(k)− x20(k) (2.33)
where xp0, x10, x20 are steady state solutions when the slow states are fixed at σ and
55
ω. |Q| and ∠Q denote the magnitude and angle of Q(z) evaluated at z = e−σc+jωc,
respectively, and σ = σ(k), ω = ω(k). By letting
x(k) =
σe(k)
ωe(k)
, y(k) =
xpe(k)
x1e(k)
x2e(k)
,
and substituting these new state variables into the system state equations, we can
derive a two time scale model form
x(k + 1) = x(k) + ε
f1(k,x,y)
f2(k,x,y)
(2.34)
y(k + 1) = A(x(k)
)y(k) (2.35)
where
f1 = −Cp(xpe(k) + xp0(k)
)(x2e(k) + x20(k)
)(x1e(k) + x10(k)
)2+(x2e(k) + x20(k)
)2 (2.36)
f2 = −KbCp(xpe(k) + xp0(k)
)(x1e(k) + x10(k)
)(x1e(k) + x10(k)
)2+(x2e(k) + x20(k)
)2 (2.37)
56
and
A =
Ap −K1Bp −K2Bp
0 exp(−σe − σc) cos(ωe + ωc) exp(−σe − σc) sin(ωe + ωc)
Cp − exp(−σe − σc) sin(ωe + ωc) exp(−σe − σc) cos(ωe + ωc)
(2.38)
For the adaptive feedback system (2.34) and (2.35), we have the following stability
and convergence theorem.
Theorem 2.1. Consider the dynamic system (2.34)-(2.35), with input signal given
by (2.24), if the following assumptions are satisfied
1. The tuning function L(z) is not equal to zero when z = e−σc+jωc, and a 6= 0;
2. For all fixed σ and ω, matrix A has eigenvalues less than one, i.e., the system
of Fig. 2.2 with H(z) given by (2.15), (2.16), has poles strictly within the unit
circle;
3. L(1) = 0,
then there exists ε∗, such that for all 0 < ε < ε∗, the origin of (2.34) and (2.35) is
locally exponentially stable. Note assumption (3 ) is not required if d(k) is zero mean,
i.e. c1 = 0.
Proof of this theorem results from direct application of Theorem 2.2.4. in [16].
This has two main requirements. It requires the fast system (2.35) to be stable,
57
which is satisfied by assumption (2), and the average of the slow system (2.34) is also
required to be stable. In [13], it is shown that
ωc2π
2π/ωc∫
0
f1dt = ω − ωc +ωcπ
× ln(σ − σc)
2 + ω2
ω2c
and
ωc2π
2π/ωc∫
0
f2dt = σ − σc +2ωcπ
tan−1 σ − σcω
where x1e, x2e and xpe are zero and time index k has been replaced by a continu-
ous time variable. Stability of the resulting average system is easily shown by Lya-
punov’s first method. The details showing that the average calculated by summation
is equivalent is more complicated and has been omitted. Other technical conditions
of Theorem 2.2.4 can be easily verified.
The two time scale requirement of the theory leads to the idea that ε∗ will be
significantly less than the magnitude of the smallest eigenvalue of (I−A). Practically,
for exponential stability of x(k) to require convergence of σ and ω to their true values
in the presence of noise, ε must be significantly greater than σc. Otherwise, x(k)
will go to zero simply as a result of d(k) going to zero. Thus selection of L(z) and ε
must be done to ensure that a three time scale system is generated in order for the
algorithm to function.
58
2.4 Simulations
In this section, the performance of the proposed adaptive algorithm is examined via
simulations. All simulations are created in MATLAB and Simulink environment using
discrete solver with time units normalized such that Ts = 1 unit time. In this case, the
frequency unit rad/s means radians per sample. For example, if a signal frequency
is 60Hz, then the Nyquist frequency is 314Hz, the sampling frequency is 628Hz,
and one sample corresponds to 1.6 milliseconds. Three signals are used to conduct
the simulations: (1) A single EDS signal with step changes on both parameters plus
a constant offset and Gaussian noise; (2) A single EDS signal with time-varying
damping factor plus a constant offset and Gaussian noise; (3) A multi-EDS modes
signal plus a constant offset and Gaussian noise. The tuning function is chosen as
L(z) = (z2− z)/(z2−0.75z+0.01), so that the closed-loop feedback system is stable.
Note that for pure sinusoidal signals, Zhang and Brown [18] presented a performance
analysis for the IMP based adaptive algorithm for uncertain frequency identification.
In this article, the tuning function L(z) is chosen as a function of ω such that the
closed-loop system is a bandpass filter with a notch of width W. By incorporating
this bandpass filter in L(z), formulae are derived for calculating the variance for the
estimated frequency. With a signal to noise ratio given by SNR, the variance of ω
was found to be ε2 ∗ W ∗ ω ∗ SNR where ε is the adaptation gain as in this work.
This analysis can be extended to EDS signals that we are interested in.
59
For the first simulation, the EDS signal is given by
d(k) = 3e−0.005k sin(0.5k) + 1 + n(k), 0 ≤ k ≤ 149.
At 150 sample point, its damping factor changes from 0.005 to 0.01 and its frequency
has a step change from 0.5 rad/s to 0.6 rad/s. In order to avoid the discontinuities
in the signal magnitude and phase, the signal is given by
d(k) = 3e−0.01k+0.75 sin(0.6k − 15) + 1 + n(k), 150 ≤ k ≤ 300.
The additive Gaussian noise n(k) has zero mean and variance 0.0001. The initial
conditions are σ(0) = 0.008, ω(0) = 0.2 rad/s. The tuning parameters are K1 =
0.5, K2 = 0.3, ε = 0.05, Kb = 2. The magnitude of the dominant eigenvalue of
matrix A is 0.9062. The algorithm presented here, with an integral action contained
in the tuning function L(z), is not affected by the presence of constant offsets, unlike
other algorithms in the literature. Fig. 2.4 shows the signal and error response plots.
The error converges to zero at 40 samples with a decay rate significantly greater
than the EDS signal’s damping factor. Fig. 2.5 shows that the IMP controller’s
parameters converge to the true values of the EDS signal at about 40 samples. After
the step changes in both parameters, it takes about 50 samples for σ and 25 samples
for ω to converge to their new values. In order to evaluate the performance of the
algorithm, we measure the variances of estimated parameters in a steady state time
60
0 50 100 150 200 250 300−2
−1
0
1
2
3
4
Sample
Sin
gle
ED
S s
igna
l
0 50 100 150 200 250 300−2
−1
0
1
2
Sample
Err
or r
espo
nse
Figure 2.4: Single EDS signal with a parameter step change and error response
period. From sample point 50 to sample point 150, the measurements are var(σ) =
7.7× 10−8, var(ω) = 1.8× 10−7.
For the second simulation, the EDS signal is
d(k) = 3 exp(∑k
i=00.02 sin(0.03i)
)sin(0.5k) + 1 + n(k),
with its damping factor defined as σc(k) = 0.02 sin(0.03k). The additive Gaus-
sian noise has zero mean and variance 0.0001. The initial conditions are σ(0) =
0.05, ω(0) = 0.2 rad/s. The tuning parameters are K1 = 0.5, K2 = 0.3, ε =
0.12, Kb = 0.83. Since the damping factor is time-varying, in order to minimize the
tracking delay, the integral gain for σ has been increased while keeping the integral
gain for ω unchanged. Fig. 2.6 shows the EDS signal with time-varying damping
61
0 50 100 150 200 250 300−0.05
0
0.05
Sample
Dam
ping
fact
or
0 50 100 150 200 250 3000.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Sample
Fre
quen
cy (
rad/
s)
Figure 2.5: Parameter estimation of a single EDS signal with a parameter step change
factor and the error response of the adaptive feedback system. It can be observed
that the error decays to zero at a rate independent of the EDS signal’s damping
factor. The top plot in Fig. 2.7 demonstrates the tracking performance of the adap-
tive algorithm. The estimated damping factor tracks the true value after 40 samples
with 8.5 samples delay. The estimated constant frequency is illustrated in the bot-
tom plot. From sample point 50 to sample point 260, the variances for estimated
parameters are calculated as var(ω) = 1.2 × 10−6 and the variance of σ defined as
var(σ(i)− 0.02 sin(0.03(i− 8.5))) equals 9.3× 10−7.
The signal for the third simulation is
d(k) = d1(k) + d2(k) + 2 + n(k),
62
0 50 100 150 200 250 300−2
−1
0
1
2
3
4
Sample
Sin
gle
ED
S s
igna
l
0 50 100 150 200 250 300−2
−1
0
1
2
Sample
Err
or r
espo
nse
Figure 2.6: Single EDS signal with time-varying damping factor and error response
0 50 100 150 200 250 300−0.05
0
0.05
Sample
Dam
ping
fact
or
Estimated σc
True σc
0 50 100 150 200 250 3000.1
0.2
0.3
0.4
0.5
0.6
Sample
Fre
quen
cy (
rad/
s)
Figure 2.7: Parameter estimation of a single EDS signal with time-varying dampingfactor
63
where
d1(k) = 2e−0.007k sin(0.3k),
d2(k) = 3e−0.012k sin(0.5k)
with initial conditions
σ1(0) = 0.005, ω1(0) = 0.2 rad/s,
σ2(0) = 0.015, ω2(0) = 0.65 rad/s.
The additive Gaussian noise has zero mean and variance 0.0001. The tuning param-
eters are the same for each IMP controller as K1 = 0.1, K2 = 0.1, ε = 0.03, Kb = 1.
The magnitude of the dominant eigenvalue of matrix A is 0.9504. Fig. 2.8 shows
the multi-EDS signal and the error response of the feedback system. As ε has been
reduced, we see slower convergence in Fig. 2.9 and 2.10. This has been seen to be
required in practice and can be inferred from the averaging proof as the averaging
period is now calculated for a sum of sinusoids. From sample point 200 to sample
point 300, the variances for estimated parameters are var(σ1) = 3.8×10−7, var(ω1) =
1.1× 10−7, var(σ2) = 4.6× 10−7, var(ω2) = 1.5× 10−7.
64
0 50 100 150 200 250 300−6
−4
−2
0
2
4
6
8
Sample
Multi−EDS signalError response
Figure 2.8: Multi-EDS signal and error response
0 50 100 150 200 250 300−0.05
0
0.05
Sample
Dam
ping
fact
or
0 50 100 150 200 250 300
0.2
0.25
0.3
0.35
Sample
Fre
quen
cy (
rad/
s)
Figure 2.9: Parameter estimation for EDS mode d1(k)
65
0 50 100 150 200 250 300−0.05
−0.04
−0.03
−0.02
−0.01
0
0.01
0.02
Sample
Dam
ping
fact
or
0 50 100 150 200 250 3000.45
0.5
0.55
0.6
0.65
Sample
Fre
quen
cy (
rad/
s)
Figure 2.10: Parameter estimation for EDS mode d2(k)
2.5 Conclusions
An IMP based adaptive algorithm is developed in discrete time for identifying expo-
nentially damped sinusoidal signals. Both the damping factor and frequency of the
signal can be estimated using the time-varying state variables of the IMP controller.
This adaptive algorithm can not only identify constant parameters, but also track
slowly time-varying parameters. By constructing a series of IMP controllers in par-
allel, the adaptive feedback system can identify a signal composed of a sum of EDS
components, with each IMP controller corresponding to one EDS component.
In the first simulation, the slow system has almost the same speed as the fast
system. In the third simulation, the fast system is faster by a factor of 2 than the slow
system. From these simulations, our algorithm has shown its functionality despite
66
the limitation that the slow system shall be slower than the fast system. In order
to speed up our algorithm, we can place the closed-loop poles closer to the origin
by tuning the function L(z). Also note that the variance measurements are zero for
noise free simulation cases.
The proposed algorithm is shown to be locally exponentially stable, with con-
vergence rates given by the design parameters, independent of the signal strength
and almost independent of the signal parameters. (By almost we refer to the natural
restrictions that convergence cannot be faster than 1/ωc, and must be faster than
σc.) Because of the local nature of the stability result, initial choice for σ and ω can
be critical.
Bibliography
[1] L. J. Brown and Q. Zhang, “Identification of Periodic Signals with Uncertain
Frequency,” IEEE Transactions on Signal Processing, vol. 51, no. 6, pp. 1538–
1545, Jun. 2003.
[2] L. A. Sievers and A. H. von Flotow, “Comparison and Extensions of Control
Methods for Narrow-band Disturbance Rejection,” IEEE Transactions on Signal
Processing, vol. 40, no. 10, pp. 2377–2391, Oct. 1992.
[3] P. A. Regalia, “An Improved Lattice-Based Adaptive IIR Notch Filter,” IEEE
Transactions on Signal Processing, vol. 39, no. 9, pp. 2124–2128, Sep. 1991.
[4] M. Bodson and S. C. Douglas, “Adaptive Algorithms for the Rejection of Sinu-
soidal Disturbances with Unknown Frequency,” Automatica, vol. 33, no. 12, pp.
2213–2221, 1997.
[5] R. Marino and P. Tomei, “Global Estimation of n Unknown Frequencies,” IEEE
Transactions on Automatic Control, vol. 47, no. 8, pp. 1324–1328, Aug. 2002.
[6] B. A. Francis and W. M. Wonham, “The Internal Model Principle of Control
Theory,” Automatica, vol. 12, pp. 457–465, 1976.
[7] T. C. Tsao, Y. X. Qian, and M. Nemani, “Repetitive Control for Asymptotic
Tracking of Periodic Signals with an Unknown Period,” Journal of Dynamic
Systems, Measurement, and Control, vol. 122, pp. 364–369, Jun. 2000.
[8] A. Serrani, A. Isidori, and L. Marconi, “Semiglobal Nonlinear Output Regula-
tion with Adaptive Internal Model,” IEEE Transactions on Automatic Control,
vol. 46, no. 8, pp. 1178–1194, Aug. 2001.
[9] R. Kumaresan and D. W. Tufts, “Estimating the Parameters of Exponentially
Damped Sinusoids and Pole-zero Modeling in Noise,” IEEE Transactions on
Acoustics, Speech, and Signal Processing, vol. ASSP-30, no. 6, pp. 833–840, Dec.
1982.
67
68
[10] R. Boyer and J. Rosier, “Iterative Method for Harmonic and Exponentially
Damped Sinusoidal Models,” in Proceedings of the 5th International Conference
on Digital Audio Effects, Hamburg, Germany, Sep.26-28, 2002, pp. 145–150.
[11] Y. Hua and T. K. Sarkar, “Matrix Pencil Method for Estimating Parameters of
Exponentially Damped/Undamped Sinusoids in Noise,” IEEE Transactions on
Acoustics, Speech, and Signal Processing, vol. 38, no. 5, pp. 814–824, May 1990.
[12] R. Badeau, R. Boyer, and B. David, “EDS Parametric Modeling and Tracking
of Audio Signals,” in Proceedings of the 5th International Conference on Digital
Audio Effects, Hamburg, Germany, Sep.26-28, 2002, pp. 1–6.
[13] J. Lu and L. J. Brown, “Control of Exponentially Damped Sinusoidal Signals,”
in Proceedings of the 2007 American Control Conference, New York City, NY,
Jul.11-13, 2007, pp. 1937–1942.
[14] H. K. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ: Prentice-Hall,
2002.
[15] S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence, and Robust-
ness. Englewood Cliffs, NJ: Prentice-Hall, 1989.
[16] E. Bai, L. Fu, and S. S. Sastry, “Averaging Analysis for Discrete Time and
Sampled Data Adaptive Systems,” IEEE Transactions on Circuits and Systems,
vol. 35, no. 2, pp. 137–148, Feb. 1988.
[17] L. J. Brown and Q. Zhang, “Periodic Disturbance Cancellation with Uncertain
Frequency,” Automatica, vol. 40, pp. 631–637, 2004.
[18] Q. Zhang and L. J. Brown, “Noise Analysis of an ALgorithm for Uncertain
Frequency Identification,” IEEE Transactions on Automatic Control, vol. 51,
no. 1, pp. 103–110, Jan. 2006.
69
Chapter 3
Combining Open and Closed-Loop
Control Via Intermittent Integral Control
Action 2
3.1 Introduction
Disturbance attenuation is an important issue in many applications. If the distur-
bance has a predictable component, it can be identified or estimated, and can be
compensated using open loop control. However, it is undesirable to use a purely open
loop controller for compensation of predictable disturbances, since slow changes, such
as drifts could eliminate the compensatory effect. In order to compensate these vari-
ations, conventional internal model principle controllers, such as integral control, are
used. But conventional internal model principle controllers limit the capabilities for
2. A version of this chapter has been submitted for publication.J. Lu and L. J. Brown, “ Combining Open and Closed-Loop Control Via IntermittentIntegral Control Action”, IEEE Transactions on Automatic Control, September, 2010
70
compensating unpredictable disturbances as a result of the phase lag they introduce
at relevant frequencies.
We propose to use internal model principle controllers whose inputs are only
connected intermittently, i.e., they switch between learned open loop and closed-loop
control. The simplest implementation of the intermittent control is the intermittent
integral control [1]. When a constant error is detected, the output error signal is
fed to the integral controller, which then learns the necessary offset to apply to the
control action. When the integral control loop is opened, the learned control action is
maintained. A standard wideband disturbance controller is used in parallel with this
switched controller to achieve desirable performance for unpredictable or non-constant
disturbances. Since the integral controller is not normally present, this wideband
controller can be made more aggressive while maintaining stability margins and/or
control actions at similar levels. Some of the work in this chapter was presented in
[1, 2], where significant performance improvements of this controller versus traditional
PI control were demonstrated.
Due to its control fashion, an intermittent integral control system can be mod-
elled as a switched system, which is defined as a hybrid dynamical system that consists
of a finite number of subsystems and a logical rule that orchestrates switching between
these subsystems [3]. Therefore, stability analysis approaches for switched systems
apply to the intermittent integral control system.
When one designs a switching scheme between multiple subsystems, a main
71
concern is that switching between these subsystems does not cause instability. There
are examples [4] demonstrating that the stability of a switched system depends not
only on the dynamics of each subsystem but also on the properties of switching
signals. It is well known that a switched system is stable if all individual subsystems
are stable and the switching is sufficiently slow, so as to allow the transient effects
to dissipate after each switch. A notable approach to formulate this idea is the dwell
time approach. This approach has been studied in [4, 5]. For switched linear systems
of which all subsystems are Hurwitz stable, Morse [6] established the fact that the
switched system is asymptotically stable provided the dwell time is chosen to be larger
than a minimum dwell time τm. This result was also extended in [7] and [8] to relax
some restrictions on each subsystem.
Another well studied approach for stability analysis with constrained switching
is based on multiple Lyapunov functions (MLF) theory. Instead of finding a common
Lyapunov function that applies for all subsystems at all time, MLF approach requires
that each subsystem has its corresponding Lyapunov function combined with a bound-
ing condition at each switching interval. These functions only require non-positive
time derivatives along the state trajectories when corresponding subsystems are ac-
tive. There are several versions of MLF results in the literature. Branicky presented
a very intuitive MLF result in [9]. He showed that when all candidate Lyapunov-like
functions for each of the individual subsystems satisfy the following conditions, a)
each function does not increase when its corresponding subsystem is active, b) each
72
function does not increase its value at each entering instant, the switched system
is stable in the sense of Lyapunov. Some other MLF approaches are discussed in
[10, 11, 12].
The intermittent integral control system belongs to a class of switched systems
that switch between controllers in the presence of constant disturbances or reference
signals or other predictable disturbances. Unless all the controllers have the same
gain at zero frequency, they will not share the same equilibrium point preventing
direct application of existing results. We extend the results in [9] by replacing the
constraint that all Lyapunov functions have negative derivatives when their corre-
sponding subsystems are active with a bounding constraint on those that do not
satisfy this stability condition. This allows some of the subsystems to not share a
common equilibrium point or even be unstable. Although the phrase “system sta-
bility” is used sometimes, strictly speaking, it refers to the stability of a system’s
equilibrium point. The origin is defined as a not stable equilibrium point if it is
either unstable or not an equilibrium point. So by expanding the classes of systems
from stable and unstable as in [8] to stable and not stable, we can allow our not stable
systems to be active for arbitrarily long. Further, all the aforementioned MLF ap-
proaches deal with switched systems that have been defined such that all subsystems
have the same states. This unnecessary condition is also dropped to include systems
that switch between controllers having different dimensions or state vectors.
This chapter is organized as follows: First, the intermittent integral control
73
system is introduced in Section 3.2. A stability theorem for switched systems based
on MLF is developed in Section 3.3. In Section 3.4, a switched system model of the
intermittent integral control system is established, followed by a stability theorem. A
numerical example of the intermittent integral control and its performance evaluation
is presented in Section 3.5, followed by the conclusion and future work in Section 3.6.
The proofs of the stability theorems are given in the appendices. The work in this
paper is based on preliminary work previously presented at several American Control
Conferences [1, 2, 13, 14].
3.2 Intermittent Integral Control System
The overall intermittent integral control system is shown in Fig. 3.1, where the
plant L(s) generally includes a standard wideband controller, such as a proportional
controller. The integral controller has a time-varying integral gain with Ki(t) =
−KdKi(t), with Kd and the opening and closing of switch S defined in Table 3.1.
(Note Ki(t) = 0 is equivalent to switch being open, and Ki(t) 6= 0 implies switch is
closed.)
The switching mechanisms of this intermittent integral controller are as follows.
Initially, the controller begins as open loop control (S open, i.e., Ki(t) = 0), and the
integral controller is initialized with a nominal offset xo(t0). The integrated error xe
is monitored. When xe exceeds a threshold xu at time t, the integral controller is
turned on by setting Ki(t) = K∗i , and xo is simultaneously augmented by xe scaled
74
S y
_
xe
L(s)r ( )iK t
s∑
e
1 s
xo
switching
mechanisms
Figure 3.1: Block diagram of an intermittent integral control system
Table 3.1: Definition of parameters for intermittent integral control
If ThenKi(t) = 0 and Ki(t
+) = K∗i ,
|xe(t)| > xu xo(t+) = xo(t) +
K∗
i xe(t)
max(1,Ks(t−tl))
0 < Ki(t) ≤ Kli Ki(t
+) = 0, xe(t) = 0, tl = t|e(t)| ≥ eu Kd = 0|e(t)| < eu Kd = Kdecay
by the time spent reaching the threshold as given by
Ki(t+) = K∗
i (3.1)
xo(t+) = xo(t) +
K∗i xe(t)
max(1, Ks(t− tl))(3.2)
where Ks is a scaling factor and the max operation guarantees no division by zero
issues. tl is the last time the integral controller is turned off or the initial time t0.
The integral controller remains active as long as the error e(t) is excessive. Once the
error is not significant, i.e., |e(t)| < eu, the integral control action is removed in a
75
smooth manner. This is achieved by setting Kd = Kdecay, the value of the integral
gain Ki(t) decays exponentially. This smooth removal of the integral action prevents
the phenomenon called chattering. When the integral gain becomes insignificant, i.e.,
Ki(t) = Kli , then Ki(t) is set to zero. Integral action is thus completely turned off
(S open). xe is reset to 0, and tl is set to t. Table 3.2 gives reasonable methods of
choosing those additional parameters listed in Table 3.1 when the design goals are
fast response and maximum disturbance attenuation.
Table 3.2: Guidelines for parameter selections
K∗i Can be chosen slightly larger than that for traditional PI control.eu Closed-loop level of noise and disturbance gives a reasonable value for eu.xu eu value multiplied by the desired response time.
Kli Between 0.1K∗
i and 0.25K∗i .
Kdecay Can be chosen to satisfy requirements of Theorem 3.2, or 0.5 to 5 timesthe reciprocal of the closed-loop rise time.
Ks Can be chosen slightly smaller than the product of K∗i and the plant
DC gain
In [1], this intermittent integral control strategy has been shown a 10% reduction
in the disturbance response than traditional PI control on plants with infrequent step
changes in set-point or load. It also remains well behaved at those operating points
where PI control system went unstable. By observing its switching mechanisms, this
control system can be modelled as a switched system. Therefore, stability analysis
approaches for switched systems can be applied to this system. However, unlike most
switched systems, the equilibrium points of this switched system cannot be guaranteed
76
to be identical or even bounded prior to establishing stability of the system.
3.3 Multiple Lyapunov Functions Based Stability
Theory
For a switched system, its switching signal σ(t) is defined as a piecewise right contin-
uous constant function of time with σ(t) ∈ {1, 2, · · · ,M}, where M is the number of
subsystems. It is assumed that there are finite number of switches in any finite time
interval. Let set {tj} represent the switching times with j being positive integer num-
bers and tj ≤ tj+1. We also define pairs of subsets of {tj} for any q ∈ {1, 2, · · · ,M}
as follows,
{tq,k} = {tj |when subsystem q is switched on};
{tq,k} = {tj |when subsystem q is switched off}
with tq,k < tq,k. Let S be a switching sequence associated with the switched system.
The interval completion I(S|q) is defined as the completion of the set of time intervals
during which subsystem q is active, i.e., I(S|q) =⋃k
[tq,k, tq,k
].
For a switched system that has different state vectors for each of the individual
subsystems, its dynamics can be described as
xq = fq(xq(t)
), t ∈ I(S|q) (3.3)
77
xq(tj) = Hq,p(xp(tj)
)(3.4)
where xq ∈ Rnq , and q, p ∈ {1, 2, · · · ,M}, (q 6= p), and σ(tj) = p, σ(tj+1) =
q. Functions Hq,p are mappings from Rnp to R
nq , and satisfy ‖Hq,p(xp(tj)
)‖ ≤
C1‖xp(tj)‖, with C1 being a constant. From (3.3), it can be seen that each state
vector is only required to be defined on its corresponding interval completion. Also
note that switching instant tj in (3.4) is both the switch-off time of subsystem p and
switch-on time of subsystem q. In this chapter, the norm of a vector x is referring
to the Euclidean norm, ‖x‖ =√xTx, and the norm of a matrix A is referring to the
induced norm, ‖A‖ = max‖x‖=1 ‖Ax‖. It is not required that each subsystem in (3.3)
has the same equilibrium point and each state vector needs only to be defined over
its corresponding interval completion.
Since each subsystem has its own state vector, the traditional definition of
Lyapunov stability needs to be modified. For dynamical system (3.3) and (3.4), when
fq(0) = 0, we say the origin of subsystem q is stable in the sense of Lyapunov if for
each ε > 0, there exist C2, δ(ε) > 0, such that if ‖xq(t0)‖ < δ, ‖xσ(t)(t)‖ < C2ε
for all t > t0, and ‖xq(t)‖ < ε for all t ∈ I(S|q), and t > t0. In addition, we can
say the origin is an asymptotically stable equilibrium point for the switched system
if ‖xσ(t)(t)‖ → 0 as t→ ∞, and the origin is stable in the sense of Lyapunov for one
subsystem q.
Since we do not require fq(0) = 0 for all q, it will not be possible to find
78
Lyapunov functions for all of the subsystems. But it is still possible to construct
energy-like functions for those not stable subsystems in a similar way as to Lyapunov
functions. These energy-like functions preserve the positive definite and continuously
differentiable properties of Lyapunov functions. But their derivatives along the solu-
tions of the system can be positive. Despite the signs of their derivatives, together
with Lyapunov functions, these energy-like functions are called candidate Lyapunov-
like functions in this paper. Note that the definition of Lyapunov function requires the
origin to be an equilibrium point, but this is not required for candidate Lyapunov-like
functions.
An MLF based theorem for the stability of switched systems:
Theorem 3.1. For a switched system described by (3.3) and (3.4), suppose we have
candidate Lyapunov-like functions Vq for each of the individual subsystems. Let Π be
the set of all switching sequences associated with the system, and Ξ be a subset of Π.
If for each S ∈ Ξ, the following conditions are satisfied,
1. There exists at least one Vi, i ∈ {1, 2, · · · ,M}, such that
(a) Vi(xi(t)) ≤ 0, for all t ∈ I(S|i);
(b) Vi(xi(ti,k+1)) ≤ Vi(xi(ti,k)), ∀k, where ti,k, ti,k+1 are two consecutive
switch-on instants of subsystem i;
2. For all other Vq’s, (q 6= i),
79
(a) xq does not have finite escape time, and is guaranteed to enter subsystem
i; or the system can be guaranteed to enter subsystem i prior to entering
subsystem q;
(b) there exists a positive constant m, such that |Vq(xq(t))| ≤ m|Vi(xi(t∗i ))|,
for tq,j ≤ t ≤ tq,j, where interval [tq,j , tq,j ] is a subset of I(S|q) for all j,
and t∗i = maxk{ti,k|ti,k < tq,j},
the switched system (3.3), (3.4) is stable in the sense of Lyapunov for all switches in
Ξ. Furthermore, if Vi(xi(t)) < 0, and one of the following two conditions is satisfied,
• the sequence {Vi(xi(ti,k))} converges to zero as k → +∞;
• {ti,k} is a finite sequence and the system stays in subsystem i after the last
switching,
the switched system is asymptotically stable.
Proof of this theorem is given in Appendix A.
3.4 Switched System Model and Stability
Theorem
Let
x = Ax+Bxo
80
y = Cx (3.5)
be a state space representation of L(s) shown in Fig. 3.1 , where x ∈ Rn. From Fig.
3.1, we have
e = r − y = r − Cx (3.6)
xo = Ki(t)e = −Ki(t)Cx+Ki(t)r (3.7)
By letting z = [xo xT ]T , z ∈ R
n+1, the intermittent integral control system can be
expressed as
z =
xo
x
=
0 −Ki(t) C
B A
xo
x
+
Ki(t)
0
r
:= Aσ(t)z +Bσ(t)r (3.8)
y =
[0 C
]xo
x
(3.9)
where Aσ(t) ∈ R(n+1)×(n+1), Bσ(t) ∈ R
(n+1)×1. The initial condition is given as
z(t0) = z0. As described in (3.2), xo is augmented when the integral controller is
turned on.
By observing the switching mechanisms, this intermittent integral control sys-
tem can be modelled as a switched linear system consisting of two subsystems. The
81
switching signal σ(t) is defined as
σ(t) =
1, if Ki(t) ∈ (Kli , K
∗i ]
2, if Ki(t) ∈ [0, Kli ]
(3.10)
Matrices Aσ(t), Bσ(t) can be written as follows,
A1(t) =
0 −K∗
i exp(∫ tt1,k
−Kddτ)C
B A
, B1(t) =
K∗i exp(
∫ tt1,k
−Kddτ)
0
(3.11)
A2(t) =
0 0
B A
, B2(t) =
0
0
(3.12)
with t ≥ t1,k in (3.11). Pair(A1(t), B1(t)
)describes the closed-loop control subsystem
with integral action, and(A2(t), B2(t)
)describes the open loop control subsystem
without integral action. For the switched linear system (3.8), (3.9), we have the
following lemma from [15],
Lemma: If the time-varying system z(t) = A1(t)z(t) satisfies the following:
1. The function t→ A1(t) is a matrix-valued piecewise continuous function bounded
on R+, i.e., supt≥0 ‖A1(t)‖ := α1 <∞;
2. There exists a positive constant µ1 such that every point-wise eigenvalue of
A1(t) satisfies Re[eigi[A1(t)]
]≤ −2µ1, ∀i, ∀t ≥ 0;
82
3. supt≥0 ‖A1(t)‖ := β satisfies β ≤ µ21/m41, where m1 = Rd(
Rd2 + α1)
2/µ31, and
Rd ≥ 2∣∣eigi[A1(t)]
∣∣, ∀i, ∀t ≥ 0,
the equilibrium of the system is uniformly exponentially stable.
To simplify the analysis, the modification of xo when subsystem 2 is switched
on is ignored by letting Ks be infinity. We then have the following theorem,
Theorem 3.2. If K∗i , K
li , and Kdecay are chosen such that the following assumptions
are satisfied,
1. The equilibrium of subsystem 1 is uniformly exponentially stable, i.e., there
exist finite positive constants γ1, λ1, such that ‖Φ1(t, τ)‖ ≤ γ1e−λ1(t−τ), where
Φ1(t, τ) is the transition matrix of subsystem 1;
2. There exists a constant µ2 > 0, such that each eigenvalue of A satisfies Re[eigi(A)] ≤
−µ2;
3. The minimum active time of subsystem 1, τ1m, given by τ1m = ln(K∗i /K
li)/Kdecay,
satisfies τ1m ≥ ln(γ21γ3
√α1/λ1
)/λ1, where γ3 is a positive constant defined in
proof,
the intermittent integral control system (3.8), (3.9) is stable in the sense of Lyapunov.
Note that assumption 1 is true if all conditions of either the Lemma or Theorem
7.4 in [16] are satisfied. The latter requires the existence of a symmetric matrix Q1(t)
with bounding conditions.
83
The proof of Theorem 3.2 is given in Appendix B.
In Fig. 3.1, the constant reference/disturbance signal r(t) is assumed to have
infrequent step changes. Let the amplitude of the first step be 1, and the second
step be Rs. The first step occurs at t0 = 0, and the second step occurs at ts. From
assumption 1 of Theorem 3.2, we have ‖z(t)‖ ≤ γ1e−λ1t, for all t ∈ I(S|1) and
t ∈ [0, ts]. After the second step, this inequality becomes ‖z(t)‖ ≤ Rsγ1eλ1(t−ts),
for all t ∈ I(S|1) and t ≥ ts. Let t1,k be the first switch-on time of subsystem 1
after the second step change of r(t). In order to maintain stability, condition (1.b) of
Theorem 3.1 needs to be modified as: ts is lower bounded such that V1(z(t1,k+1)
)≤
V1(z(t1,k−1)
). The lower bound of ts is a function of Rs.
3.5 A Numerical Example
In order to demonstrate the performance of the intermittent integral control system,
a numerical example is given below that shows the benefits achieved relative to simple
feedback control and a state feedback control with an augmented integrator which is
presented in [17]. We consider a simple second order plant given by
xp =
−0.8 −1
1 0
xp +
1
0
u (3.13)
y =
[13 1
]xp (3.14)
84
Step responses of systems with state feedback control, state feedback control with
augmented integrator, and the intermittent integral control are simulated. In order to
understand both steady state behaviour and transient behaviour, the reference signal
r(t) is chosen to have a step change from 0 to 1 at t = 50, which is the midpoint of
the simulation. A white Gaussian noise n(t) with zero mean and standard deviation
σ = 0.2 is added as the wideband disturbance to the system.
To evaluate the benefits of intermittent integral control, the controller design
is formulated as stochastic optimal linear regulator problems with the presence of
wideband disturbances. The optimization criterion is defined as follows,
J = E{
limtf→∞
tf∫
t0
[y2(t) + uT (t)Ru(t)]dt}
(3.15)
where R is a weighting matrix.
3.5.1 Controller 1
By implementing linear quadratic regulator (LQR) design method, the optimal state
feedback controller gain is [18]
K = −R−1BTp P, (3.16)
85
where P is the solution to the algebraic Riccati equation (ARE),
ATp P + PAp − P (BpR
−1BTp )P + CT
p Cp = 0 (3.17)
and (Ap, Bp, Cp) is a state representation of the plant. A smaller weighting ofR results
in larger absolute values of K, which gives faster response and smaller overshoot. But
the trade-off is a greater variance in the steady state error. With the choice of R = 0.5,
the optimal state feedback gain for controller 1 is K = [−0.7252, −0.7321].
3.5.2 Controller 2
An augmentation method is implemented for the design of an optimal state feedback
controller with integral action. An additional state xo, given as the integral of the
difference between the output and reference signal, is added to the plant. The LQR
design is then applied to the augmented system to obtain an optimal state feedback
gain Ke. Note the third value of this vector is the integral gain K∗i . For design
purposes only, the optimization criterion is J = E{∫∞t0
(y2 + ρx2o +Ru2)dt}, where ρ
is the weighting on the state xo. When the value of ρ is zero, the augmented system
is not observable. As ρ approaches zero, the controller converges to the proportional
controller and step response becomes unacceptably slow. With the choice of ρ = 0.4,
which is explained in next subsection, the optimal state feedback gain for controller
2 is Ke = [−1.2079, −1.2866, 0.8944], and K∗i = 0.8944.
86
3.5.3 Intermittent integral controller
In this example, L(s) in Fig. 3.1 is designed as the combination of the plant and
controller 1 obtained earlier. K∗i is the gain for the augmented state xo obtained
from controller 2 above. Since the response with controller 1 has standard deviation
σ1 = 0.005 and rise time tr = 1.24, according to the guidelines in Table 3.2, we choose
the parameters of the intermittent integral controller as follows, eu = 4σ1 = 0.02,
xu = eutr = 0.025, Ks = 0.8, Kli = 0.1 ∗K∗
i , and Kdecay = 0.25. From hereinafter,
intermittent integral controller is also referred to as controller 3.
In order to verify the assumptions of Theorem 3.2, two positive definite sym-
metric matrices Q1 and Q2, corresponding to the two subsystems, are calculated by
solving corresponding Lyapunov equations. According to Theorem 7.4 in [16], the
existence of these two matrices can guarantee assumptions 1 and 2. For subsystem
1, we obtain Q1 by solving the Lyapunov equation AT1 (t)Q1 + Q1A1(t) = −I at
t = 0, i.e., Ki(t) = K∗i . For all Ki(t) ∈ [Kl
i , K∗i ], we verified that Q1 satisfies
AT1 (t)Q1 + Q1A1(t) ≤ −νI, where ν is a positive constant. Since matrix A is time
invariant, Q2 can be simply obtained by solving equation ATQ2 +Q2A = −I.
Q1 =
1.9784 −0.5 −1.1907
−0.5 1.018 0.9035
−1.1907 0.9035 2.3391
, Q2 =
0.5171 0.2887
0.2887 1.3359
The existence of these two matrices justifies the first two assumptions of Theorem
87
3.2. Consequently, following parameter values from Appendix B can be calculated,
λ1 = 0.5, γ1 = 2, α1 = 2.7115, µ2 = 0.7626, γ2 = 1.4755, α2 = 2.4104, and
γ3 = 2.9133. Thus the lower bound of the minimum active time of subsystem 1 is
ln(γ21γ3√α1/λ1)/λ1 = 6.6018. The minimum active time is τ1m = ln(K∗
i /Kli)/Kdecay =
9.2103 > 6.6018. Assumption 3 of Theorem 3.2 is satisfied.
Cost functions are calculated using the following formula for different values of
ρ and shown in Table 3.3,
J =
tf∫
0
(e2(t) + u2(t)R)dt (3.18)
where tf = 100. To distinguish between the transient performance and noise rejection
capability of the approach, the cost is broken into two time periods, 0 ≤ t ≤ 50 where
the cost Js is driven by the additive noise, and 50 < t ≤ 100 where the cost Jt
is primarily a function of the step response. It can be seen that as ρ increases,
for controller 2, Js increases and Jt decreases. To achieve better constant reference
tracking, the integral controller must trade off its wideband disturbance attenuation
performance. For controller 3, since the optimal state feedback gain K, which is
independent of ρ, does not change, Js remains unchanged. However, as ρ increases,
Jt decreases first and increases after reaching its minimum value at ρ = 0.4.
To illustrate the step response performances of the controllers, we choose ρ = 0.4
for the following results. Thus the state feedback gain and integral gain for controller
88
Table 3.3: Cost function values of controller 2 & controller 3’s step responses
ρ K∗i controller 2 controller 3
Js Jt Js Jt0.2 0.6325 0.00222 0.9738 0.00186 0.74950.3 0.7746 0.00230 0.9072 0.00186 0.70550.4 0.8944 0.00238 0.8668 0.00186 0.69120.6 1.0954 0.00250 0.8177 0.00186 0.69910.7 1.1832 0.00256 0.8010 0.00186 0.71211.0 1.4142 0.00271 0.7659 0.00186 0.7711
3 are K = [−0.7252, −0.7321], and K∗i = 0.8944. Fig. 3.2 shows the intermittently
invoked integral control action of controller 3. After the first switch-on, the integral
control is at its full strength with maximum integral gain K∗i because the tracking
error is excessive. When the tracking error decreases below the threshold eu, the
integral gain starts to decay exponentially. When Ki(t) decays to about 0.86 at
t = 52, the effects of the noise cause |e(t)| to be greater than eu again. According
to the switching mechanisms defined in Table 3.1, Kd = 0 when |e(t)| > eu, which
means Ki(t) stops decaying and keeps its value as it is. At about t = 54, the condition
|e(t)| < eu is satisfied again, Ki(t) continues its decaying and stops decreasing again
between t = 54.4 and t = 56.9. It is then set to zero when the lower bound Kli is
reached. This process is also shown in the inset plot. Note there remains a small
residual error in the system that results in the integrator again being invoked at
about t = 67. The integral gain immediately starts to decrease because the tracking
error has already decreased below the threshold. There remains an exponentially
decreasing residual error between the set point and the system output that continues
89
to cause the integral action to be invoked. However, the time between invocations
increases exponentially.
0 10 20 30 40 50 60 70 80 90 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time
Inte
gral
gai
nIntermittent integral control action
52 54 56 580.6
0.65
0.7
0.75
0.8
0.85
0.9
Figure 3.2: Integral gain of the intermittent integral control system
The top plot in Fig. 3.3 shows the tracking errors of the three controllers’ step
responses. The bottom plot shows the control signals during the transient period.
The steady state control signals of controller 2 and controller 3 are plotted in Fig.
3.4. The variance of controller 3’s control signal is 1.32× 10−5, which is a significant
improvement than that of controller 2 (3.46× 10−5).
Compared with the costs of controller 2 and 3 for ρ = 0.4, the costs of con-
troller 1 is Js = 0.00186, and Jt = 9.3588. As we can see, in steady state, when
90
0 10 20 30 40 50 60 70 80 90 100−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Time
Tra
ckin
g er
ror
Step response
50 52 54 56 58 60 62
0
0.2
0.4
0.6
0.8
1
Time
Con
trol
sig
nal
Controller 3Controller 1Controller 2
Figure 3.3: Tracking errors and control signals of the three control systems
the parameter eu is larger than the steady state noise, performance equivalent to
optimal control without integral action is achieved. Further the switching mechanism
successfully allows the system to perfectly reject constant disturbances and perfectly
follow infrequent step changes in set points with minimal increase in costs during the
transient.
3.6 Conclusions
A new control strategy that combines the benefits of closed-loop control with open
loop control of predictable disturbances is presented. A tool that can be used to test
91
60 65 70 75 80 85 90 95 1000.97
0.98
0.99
1
1.01
1.02
1.03
Time
Con
trol
ler
2
Steady state control signal
60 65 70 75 80 85 90 95 1000.97
0.98
0.99
1
1.01
1.02
1.03
Time
Con
trol
ler
3
Figure 3.4: Comparison of steady state control signals of controller 2 & controller 3
for stability of the resulting system is also presented. This tool can then be used to
assist in selecting the switching parameters. Due to the intermittent control fashion
of the integral action, a feedback system with the intermittent integral controller
can be modelled as a switched system with two constituent systems. Since the two
subsystems do not share common equilibrium point, and have different state variables,
existing approaches can not be applied to the intermittent integral control system
directly. By extending the MLF result in [9], a new MLF theorem is developed
with relaxations on the constraints on constituent systems. The intermittent integral
control system is proved to be stable in the sense of Lyapunov by applying this new
92
MLF theorem. A numerical example shows that the intermittent integral control
system improves the reference tracking and disturbance rejection performance than
traditional state feedback control systems with or without integral control.
This intermittent integral control approach is capable of cancelling constant
disturbances and attenuating unpredictable disturbances. But for systems subject
to narrowband disturbances and unpredictable disturbances, this approach has its
limitations. In [2], the concept of integral control was extended to deal with any
predictable disturbances using Internal Model Principle (IMP) controller. This con-
trol strategy is known as Intermittent Cancellation Control (ICC). However, ICC is
restricted to plants that have infrequent changes in set point or disturbance charac-
teristics, and the period of the predictable disturbance is required a priori. As of our
future work, we will extend the IMP controller in the ICC approach to an adaptive
IMP control algorithm, developed by Brown and Zhang [19]. The adaptive IMP feed-
back loop is invoked to identify and cancel the predictable disturbances. Once the
predictable disturbances are cancelled, the feedback loop is opened creating an open
loop controller. The main challenge will be finding a switching sequence to stabilize
the switched system.
Appendix A Proof of Theorem 3.1
Proof : The proof follows directly the proof of Theorem 2.3 in [9]. We do the proof
for the case of M = 2. It can be easily extended to the case of M > 2.
93
Assume V1(x1(t)), V2(x2(t)) are two candidate Lyapunov-like functions cor-
responding to subsystem 1 and 2 respectively. x1 and x2 are corresponding state
vectors. V1(x1(t)) satisfies condition 1 of the theorem, and V2(x2(t)) satisfies condi-
tion 2. There is no requirement on the sign of time derivative of V2(x2(t)). As shown
in Fig. 3.5, V2(x2(t)) may increase or decrease its value during active time intervals.
In this figure, ti’s are switching instants. If V2(x2(t)) ≤ 0, this theorem is equivalent
to Theorem 2.3 in [9]. We consider the worst case of V2(x2(t)) > 0 in the proof.
At initial time, either subsystem could be active. If subsystem 2 is active initially,
V2(x2(t)) can only reach a finite value when subsystem 1 is switched on in finite time
because of condition (2.b).
mV1(t1)
V1(x1):V2(x2):
t0 t1 t2 t3 t4 t5 t6 t7 t8 t9
mV1(t3)
mV1(t5)
mV1(t7)
Vq(xq)
t
Figure 3.5: Multiple Lyapunov functions stability
Let Sp(r), B(r) represent the sphere, and ball of Euclidean radius r about the
origin respectively. Let mini(r) denote the minimum value of Vi(xi(t)) on Sp(r).
During time interval [t1, t2], subsystem 1 is active. For any given R1, we can pick a
94
r1 ∈ (0, R1), such that in B(r1), we have V1(x1(t)) < min1(R1). This is achievable
via the continuity of V1(x1(t)). Then if x1(t1) starts in B(r1), x1(t) will stay in B(R1)
during this interval.
At t2, subsystem 2 is switched on. Since V2(x2(t)) is upper bounded by
mV1(x1(t1)) from condition (2.a), we let this bound be the minimum value of V2(x2(t))
on a sphere of radius R21, i.e., min2(R21) = mV1(x1(t1)). Then during time interval
[t2, t3], x2(t) will move away from the origin, but is confined within B(R21).
From condition (1.b), we know that V1(x1(t3)) ≤ V1(x1(t1)), which means if
x1(t1) starts in B(r1), x1(t3) starts in B(r1). Then x1(t) stays in B(R1) during
interval [t3, t4]. Therefore, if x1(t1) starts in B(r1), x1(t) will stay in B(R1) during
every time interval when subsystem 1 is active.
Because V1(x1(t3)) ≤ V1(x1(t1)), the upper bounds on V2(x2(t)) also have
the relationship mV1(x1(t3)) ≤ mV1(x1(t1)). If we let min2(R22) = mV1(x1(t3)),
the radius R22 of B(R22) is no larger than R21. Then during time intervals when
subsystem 2 is active, x2(t) is always confined within B(R21).
As depicted in Fig. 3.6, for any given R1, we can always find a r1 and a R21,
such that if x1(t) starts in B(r1), x1(t) stays in B(R1) when subsystem 1 is active,
and x2(t) stays in B(R21) when subsystem 2 is active. This proves that the switched
system is stable in the sense of Lyapunov.
If V1(x1(t)) < 0, when subsystem 1 is active, x1(t) will approach its equilibrium
point at the origin as time elapses. First, if the switching sequence {tj} is infinite,
95
r1
x1
x2
R21R1
Figure 3.6: Switched system with different equilibria
and the sequence {V1(x1(t1)), V1(x1(t3)), · · · , V1(x1(t1,k)), · · · } converges to zero as
k → ∞, x1(t) approaches the origin. The upper bounds on V2(x2(t)) also converges
to zero under this condition. Thus x2(t) is confined within a ball that is getting
smaller and smaller every time subsystem 2 is active, and eventually, this ball will
shrink to the origin as k → ∞. Therefore, state trajectory x1(t) approaches its
equilibrium points at the origin, and x2(t) approaches the origin as well. Second, if
the sequence {t1, t3, t5, · · · , t1,k, · · · } is finite and after the last switching, the system
stays in subsystem 1, which is asymptotically stable, x1(t), which is now the state
of the switched system, will approach its equilibrium point at the origin. In both
circumstances, the switched system’s state trajectories approach the origin. This
proves the asymptotic stability of the switched system.
Appendix B Proof of Theorem 3.2
Proof : By studying the structure of the system matrices (3.11) and (3.12), Theorem
3.1 developed in Section 3.3 can be applied for the stability analysis of system (3.8),
96
(3.9). In order to apply Theorem 3.1, each subsystem is analysed so that correspond-
ing candidate Lyapunov-like functions can be constructed.
Analysis of Subsystem 1
It can be seen from (3.11) to (3.12) that subsystem 1 is linear time-varying (LTV),
while subsystem 2 is linear time invariant (LTI). The equilibrium of subsystem 1 is
at the origin.
Since all entries in A1(t) are bounded for all t > 0, there exists α1 > 0 such
that ‖A1(t)‖ ≤ α1. From Theorem 7.8 in [16], matrix
Q1(t) =
∞∫
t
ΦT1 (σ, t)Φ1(σ, t)dσ (3.19)
satisfies
1
2α1≤ ‖Q1(t)‖ ≤ γ21
2λ1(3.20)
We can pick a candidate Lyapunov-like function for subsystem 1 as V1(z(t)
)=
zTQ1(t)z. Then we have
1
2α1‖z‖2 ≤ V1(z(t)) ≤
γ212λ1
‖z‖2 (3.21)
97
and
V1 = zT [AT1 (t)Q1(t) + Q1(t)A1(t)]z + zT Q1(t)z = −‖z‖2 (3.22)
Analysis of Subsystem 2
The state-space equations of subsystem 2 are
xo = 0 (3.23)
x = Ax+Bxo (3.24)
Since xo remains constant when subsystem 2 is active, subsystem 2 can be considered
as a forced linear system with reduced number of states. As expressed in (3.24), the
state xo in subsystem 1 can be seen as a constant input to (3.24). The number of
states for (3.24) is n, while subsystem 1 has (n+ 1) states.
Since Re[eigi(A)] ≤ −µ2 from assumption 2, the state transition matrix of A
satisfies
‖eA(t−τ)‖ ≤ γ2e−µ2(t−τ) (3.25)
98
where constant γ2 can be derived following the Lemma in [20]. For n× 1 matrix
Q2 =
∞∫
0
eAT τ eAτdτ, (3.26)
it satisfies
1
2α2≤ ‖Q2‖ ≤ γ22
2µ2(3.27)
where α2 is an upper bound of ‖A‖. We can pick a candidate Lyapunov-like function
for subsystem 2, V2(x(t)) = xTQ2x. Then we have
1
2α2‖x(t)‖2 ≤ V2(x(t)) ≤
γ222µ2
‖x(t)‖2 (3.28)
The complete solution of (3.24) is
x(t) = eA(t−t2,k)x(t2,k) +
t∫
t2,k
eA(t−σ)Bxodσ (3.29)
with switch-on time at t2,k. Therefore, for all t, τ , such that t2,k ≤ τ < t ≤ t2,k,
xo(t)
x(t)
=
1 0
∫ tτ e
A(t−σ)Bdσ eA(t−τ)
xo(τ)
x(τ)
:= eA2(t−τ)
xo(τ)
x(τ)
(3.30)
Let w be partitioned as [ws, wTv ]
T , where ws is a scalar, and wv is an n × 1 vector.
99
We have
eA2(t−τ)w =
ws
∫ tτ e
A(t−σ)Bdσws + eA(t−τ)wv
(3.31)
Thus
‖eA2(t−τ)w‖ =
√√√√√w2s + ‖
t∫
τ
eA(t−σ)Bdσws + eA(t−τ)wv‖2
≤
√√√√√w2s + 2‖
t∫
τ
eA(t−σ)Bdσ‖2w2s + 2‖eA(t−τ)wv‖2 (3.32)
Let wvm be the vector that maximizes ‖eA(t−τ)wv‖, i.e.,
‖eA(t−τ)‖ = max‖wv‖=1
‖eA(t−τ)wv‖ = ‖eA(t−τ)wvm‖ (3.33)
Clearly, vector w that maximizes (3.32) is of the form
w =
α
√1− α2wvm
, (0 ≤ α ≤ 1) (3.34)
Since B is known, and from (3.25), we have
‖t∫
τ
eA(t−σ)Bdσ‖ ≤t∫
τ
‖eA(t−σ)‖dσ‖B‖
100
≤t∫
τ
γ2e−µ2(t−σ)dσ‖B‖
=γ2µ2
(1− e−µ2(t−τ))‖B‖
≤ γ2µ2
‖B‖ (3.35)
Inequality (3.32) can be further simplified as
‖eA2(t−τ)w‖ ≤√(
1 +2γ22µ22
‖B‖2)α2 + 2(1− α2)γ22 (3.36)
‖eA2(t−τ)w‖ takes maximum value when α = 0 or 1. Thus
‖eA2(t−τ)‖ ≤ max{√
1 +2γ22µ22
‖B‖2,√2γ2
}:= γ3 (3.37)
Analysis of the Overall Control System
For switched system (3.8), its switching signal σ(t) ∈ {1, 2}. {A1, A2} in (3.11)-
(3.12) constitute a family of matrices describing the subsystems. As stated in Section
3.2, initially, the switched system begins with open loop control, i.e., subsystem 2
is active initially. Let N be the number of switches over the interval (t0, t) with set
{t1, t2, · · · , tN} representing the switching times. The switching on and off time sets
for the two subsystems are
{t1,k} = {t1, t3, · · · , tN}; {t1,k} = {t2, t4, · · · , tN−1}; (3.38)
101
{t2,k} = {t2, t4, · · · , tN−1}; {t2,k} = {t3, t5, · · · , tN}. (3.39)
The interval completions for the two subsystems are
I (S|1) = [t1, t2]⋃
[t3, t4]⋃
· · ·⋃
[tN , t] (3.40)
I (S|2) = [t0, t1]⋃
[t2, t3]⋃
· · ·⋃
[tN−1, tN ] (3.41)
Note that the initial time t0 and final time t are not considered as switching times,
but they are included in the interval completions.
In Subsection 3.6, candidate Lyapunov-like function V1(z(t)) has been shown to
satisfy condition (1.a) of Theorem 3.1 in (3.22). Since there are only two subsystems
and the switching sequence is minimal, for any four consecutive switching times,
tk−1, tk, tk+1, tk+2,
z(tk+2) = eA2(tk+2−tk+1)z(tk+1) = eA2(tk+2−tk+1)Φ1(tk+1, tk)z(tk) (3.42)
where it is assumed that subsystem 1 is switched on at tk and tk+2, and subsystem
2 is switched on at tk−1 and tk+1.
From assumption 1 of Theorem 3.2 and (3.37), we have
‖z(tk+2)‖ ≤ ‖eA2(tk+2−tk+1)‖ · ‖Φ1(tk+1, tk)‖ · ‖z(tk)‖ ≤ γ1γ3e−λ1τk‖z(tk)‖
(3.43)
102
where τk = tk+1 − tk is the active time of subsystem 1 between the two consecutive
switches.
Since subsystem 1 switches to subsystem 2 when Ki(t) decays to its lower bound
Kli , the minimum active time for subsystem 1 can be obtained as,
τ1m =1
KdlnK∗i
Kli
(3.44)
Considering assumption 3 of Theorem 3.2, which can also be written as
γ21γ3√2λ1
e−λ1τ1m ≤ 1√2α1
(3.45)
Inequality (3.43) is then equivalent to
γ1√2λ1
‖z(tk+2)‖ ≤ γ21γ3√2λ1
e−λ1τ1m‖z(tk)‖ ≤ 1√2α1
‖z(tk)‖ (3.46)
or
γ212λ1
‖z(tk+2)‖2 ≤ 1
2α1‖z(tk)‖2 (3.47)
Considering the upper and lower bounds on V1(z(t)) in (3.21), we have
V1(z(tk+2)
)≤ γ21
2λ1‖z(tk+2)‖2 ≤ 1
2α1‖z(tk)‖2 ≤ V1
(z(tk)
)(3.48)
103
which means that at tk+2, V1 is no greater than its value last time when subsystem
1 is switched on at tk. Thus condition (1.b) of Theorem 3.1 is verified.
At tk+1,
‖z(tk+1)‖2 = ‖xo(tk+1)‖2 + ‖x(tk+1)‖2 ≥ ‖x(tk+1)‖2 (3.49)
During interval [tk+1, tk+2], subsystem 2 is active, so for all t ∈ [tk+1, tk+2]
‖xo(t)‖ = ‖xo(tk+1)‖
‖x(t)‖ ≤ γ2e−µ2(t−tk+1)‖x(tk+1)‖
and from (3.28),
V2(x(t)) ≤γ222µ2
‖x(t)‖2 ≤ γ422µ2
exp(−2µ2(t− tk+1))‖x(tk+1)‖2 ≤ γ422µ2
‖x(tk+1)‖2
(3.50)
From (3.49) and (3.50), we have
V1(z(tk+1)) ≥1
2α1‖z(tk+1)‖2 ≥ 1
2α1‖x(tk+1)‖2 ≥ µ2
α1γ42
V2(x(t)) (3.51)
Since during [tk, tk+1], V1 < 0, we have V1(z(tk)) ≥ V1(z(tk+1)), which gives
V2(x(t)) ≤α1γ
42
µ2V1(z(tk)) (3.52)
104
for all t ∈ [tk+1, tk+2].
All conditions of Theorem 3.1 are thus satisfied. Therefore, the switched system
(3.8) is stable in the sense of Lyapunov, which proves Theorem 3.2.
Bibliography
[1] L. J. Brown, G. E. Gonye, and J. S. Schwaber, “Non-linear PI Control Inspired
by Biological Control Systems,” in Proceedings of the 37th IEEE Conference on
Decision and Control, Tampa, FL, Dec. 1998, pp. 1040–1045.
[2] L. J. Brown and J. S. Schwaber, “Intermittent Cancellation Control: A Control
Paradigm Inspired by Mammalian Blood Pressure Control,” in Proceedings of
the American Control Conference, San Diego, CA, Jun. 1999, pp. 139–143.
[3] A. V. Savkin and R. J. Evans, Hybrid Dynamical Systems: Controller and Sensor
Switching Problems. Boston, MA: Birkhauser, 2002.
[4] R. A. DeCarlo, M. S. Branicky, S. Pettersson, and B. Lennartson, “Perspectives
and Results on the Stability and Stabilizability of Hybrid Systems,” Proceedings
of the IEEE, vol. 88, no. 7, pp. 1069–1082, Jul. 2000.
[5] H. Ye, A. N. Michel, and L. Hou, “Stability Theory for Hybrid Dynamical Sys-
tems,” IEEE Transactions on Automatic Control, vol. 43, no. 4, pp. 461–474,
Apr. 1998.
[6] A. S. Morse, “Supervisory Control of Families of Linear Set-Point Controllers–
Part 1: Exact Matching,” IEEE Transactions on Automatic Control, vol. 41,
no. 10, pp. 1413–1431, Oct. 1996.
[7] J. P. Hespanha and A. S. Morse, “Stability of Switched Systems with Aver-
age Dwell-time,” in Proceedings of the 38th IEEE Conference on Decision and
Control, Phoenix, AZ, Dec. 1999, pp. 2655–2660.
[8] G. Zhai, B. Hu, K. Yasuda, and A. N. Michel, “Stability Analysis of Switched
Systems with Stable and Unstable Subsystems: An Average Dwell Time Ap-
proach,” in Proceedings of the American Control Conference, Chicago, IL, Jun.
2000, pp. 200–204.
105
106
[9] M. S. Branicky, “Multiple Lyapunov Functions and Other Analysis Tools for
Switched and Hybrid Systems,” IEEE Transactions on Automatic Control,
vol. 43, no. 4, pp. 475–482, Apr. 1998.
[10] D. Liberzon, Switching in Systems and Control. Boston, MA: Birkhauser, 2003.
[11] G. Zhai, I. Matsune, J. Imae, and T. Kobayashi, “A Note on Multiple Lya-
punov Functions and Stability Condition for Switched and Hybrid Systems,” in
Proceedings of the 16th IEEE International Conference on Control Applications,
Singapore, Oct.1-3, 2007, pp. 226–231.
[12] H. Ye, A. N. Michel, and L. Hou, “Stability Analysis of Discontinuous Dynamical
Systems with Applications,” in Proceedings of the 13th World Congress of the
International Federation of Automatic Control, vol. E, San Francisco, CA, Jun.
1996, pp. 461–466.
[13] J. Lu and L. J. Brown, “A Multiple Lyapunov Functions Approach for Stability
of Switched Systems,” in Proceedings of the 2010 American Control Conference,
Baltimore, MD, Jul. 2010, pp. 3253–3256.
[14] ——, “Stability Analysis of a Proportional with Intermittent Integral Control
System,” in Proceedings of the 2010 American Control Conference, Baltimore,
MD, Jul. 2010, pp. 3257–3262.
[15] C. A. Desoer, “Slowly Varying System x = A(t)x,” IEEE Transactions on Au-
tomatic Control, pp. 780–781, Dec. 1969.
[16] W. J. Rugh, Linear System Theory, 2nd ed. Upper Saddle River, NJ: Prentice-
Hall, 1996.
[17] C.-T. Chen, Linear System Theory And Design, 3rd ed. New York, NY: Oxford
University Press, 1999.
[18] H. Kwakernaak and R. Sivan, Linear Optimal Control Systems. John Wiley
and Sons, Inc., 1972.
[19] L. J. Brown and Q. Zhang, “Periodic Disturbance Cancellation with Uncertain
Frequency,” Automatica, vol. 40, pp. 631–637, 2004.
107
[20] B. Barkat, “Stability of linear time-varying systems,” in Proceedings of Interna-
tional Symposium on Signal Processing and its Applications, Gold Coast, Aus-
tralia, Aug.25-30, 1996, pp. 246–249.
108
Chapter 4
A Combination of Open and Closed-loop
Control for Disturbance Rejection 3
4.1 Introduction
Disturbance rejection is a major control issue in control systems which has been
studied for several decades. In general, disturbances can be categorized into two
types: predictable and unpredictable. Predictable disturbances are also referred to as
narrowband disturbances, which include sinusoidal or sum of sinusoidal signals. Un-
predictable disturbances include white or colored noises which have wide bandwidth
in frequency domain. For these two types of disturbances, two different controllers
can be designed with their own attempts to achieve best performance.
A popular technique for predictable disturbance rejection is based on the In-
ternal Model Principle (IMP) proposed by Francis and Wonham in 1976. The main
3. A version of this chapter has been submitted for publication.J. Lu and L. J. Brown, “ A Combination of Open and Closed-loop Control for DisturbanceRejection”, 24th Canadian Conference on Electrical and Computer Engineering, December,2010
109
idea of IMP is that a suitably reduplicated model of the dynamic structure of the
disturbance or reference should be incorporated in a stable feedback loop for perfect
disturbance cancellation or reference tracking. The purpose of the IMP controller is
to supply right-half plane closed-loop transmission zeros to cancel the unstable poles
of exogenous signals [1]. The IMP approach requires the knowledge of the disturbance
a priori. The accuracy of regulation depends on the fidelity of the IMP controller.
If the frequencies of the predictable signals drift over time or are unknown,
the traditional IMP approach cannot achieve perfect or possibly acceptable rejection.
There are many algorithms that can identify or estimate unknown parameters of
predictable disturbances to achieve perfect rejection. These algorithms include higher
harmonic control [2], LMS adaptive feedforward filtering [3], adaptive feedforward
cancellation (AFC) [4], and adaptive IIR notch filter [5].
In [6, 7], an IMP based adaptive algorithm was developed to identify and cancel
quasi-periodic or narrowband disturbances with uncertain frequencies. According to
the IMP, for a sinusoidal disturbance with frequency ωc, the IMP controller must
have a pair of marginally stable poles at s = ±jωc. Since ωc is unknown, a standard
IMP controller is implemented through a block diagonal state space model, in which
the best estimate of the frequency is used. A simple mapping from the states of
the IMP controller to the errors in the frequency estimates was developed. This
“measurement” of the frequency error is then used in the adaptation of the IMP
controller parameters, so that these parameters converge to their true values. When
110
this adaptive IMP controller is placed into a feedback loop, the resulting closed-loop
system produces zero steady state error for sinusoidal disturbances with unknown
frequencies. Periodic or quasi-periodic disturbances that consist of multiple sinusoidal
components, can also be perfectly cancelled by placing a number of these adaptive
modules in parallel. Since adaptation is always present, this method is able to track
slow variations in frequency, causing it to have almost perfect rejection for narrowband
and sums of narrowband signals.
However, when a system is subject to both predictable and unpredictable dis-
turbances, the presence of an IMP controller introduces a phase lag at relevant fre-
quencies which will limit the wideband disturbance controller’s capabilities for com-
pensating unpredictable disturbance. This limitation can be resolved by using the
intermittent cancellation control technique introduced by Brown et al. in [8, 9]. A
simple implementation of this techniques is introduced in [10], where the predictable
disturbance is constant. The constant disturbance is rejected using an integral con-
troller, which is one of the simplest IMP controllers. When the error between output
and disturbance signal is excessive, the integral control action is fully engaged with
the maximum gain by design. When the error is not significant, the input to the
integral controller is removed and the control action is maintained. A standard wide-
band disturbance controller is used to achieve desirable performance for unpredictable
disturbances. Since the integral controller is not always present, this wideband dis-
turbance controller can be made more aggressive while maintaining stability margins
111
and/or control actions at similar levels.
As an extension to the intermittent integral control, this chapter considers peri-
odic signals with unknown frequencies as predictable disturbance. An adaptive IMP
controller will be used to identify the unknown frequency and cancel the sinusoid. The
input to the IMP controller is connected intermittently as to allow a wideband dis-
turbance controller to minimize the unpredictable disturbance more efficiently. This
chapter is organized as follows. The intermittent cancellation control strategy is dis-
cussed in Section 4.2. A switched system model of the intermittent control is derived
followed by a stability theorem in Section 4.3. Numerical examples are presented in
Section 4.4, followed by some concluding remarks in Section 4.5.
4.2 Open and Closed-Loop Control Strategy
The block diagram of the proposed control system is shown in Fig. 4.1. The block
L(s) generally includes both the process to be controlled with a standard wideband
disturbance controller. The sinusoidal disturbance is r(t) = a sin(ωct + φ), where
its parameters a, ωc, φ are all unknown. The unpredictable disturbance n(t) is an
additive white Gaussian noise.
According to [6], a state space representation of the adaptive IMP controller
112
!
"!#"#!#$% &%
'
_
(
$%&'()&*+#
,-().*&/,/
)!#"#!
!
*%
Figure 4.1: Block diagram of open and closed-loop control system
M(s) is given by,
x1
x2
=
0 ω
−ω 0
x1
x2
+
0
f(t)
e (4.1)
u =
[K1 K2
]x1
x2
(4.2)
ω =
Kωex1
x21+x2
2
, if (x21 + x22) ≥ xu
0, if (x21 + x22) < xu
(4.3)
where K1, K2 are two tuning gains of the IMP controller. Normally, the scalar
function f(t) takes its value at 0 or 1, where the 0 state represents the switch S
in Fig. 4.1 being open. Although the true frequency of the sinusoid ωc is unknown,
according to the certainty equivalence principle, it can be replaced by its best estimate
ω as shown in (4.1). This estimated frequency ω is then updated in (4.3) with an
113
adaptation gain Kω, and is proved to converge to the true frequency in [6]. In order
to avoid noise issues related to dividing a small number by another small number,
the frequency adaptation is stopped when the squared norm of the two states is less
than a threshold xu.
The tracking error e(t) is the input to the IMP controller, and is connected
intermittently with the opening and closing of the switch S. The switching control
of S is based on two monitored signals: e(t) and xm(t), where xm =√x2m1 + x2m2,
and xm1, xm2 are the states of a monitored internal model given by
xm1
xm2
=
0 ω
−ω 0
xm1
xm2
+
0
1
e (4.4)
Correspondingly, there are two predefined thresholds eu and xu for the switching
control.
The system begins as open loop control (S open, i.e., f(t) = 0) with the
adaptive IMP controller being initialized. When the monitored signal xm exceeds the
threshold xu at time t, the closed-loop control is switched on by setting f(t) to its
maximum value 1. The states of the IMP controller are also modified,
f(t+) = 1 (4.5)
xj(t+) = xj(t) +
xm,j(t)
Ks(t− tl), j ∈ {1, 2} (4.6)
114
where Ks is a scaling factor, and tl is the last time the IMP controller is turned off or
the initial time t0. In (4.6), the IMP controller’s states are augmented by the states of
monitored internal model which are scaled by the time spent reaching the threshold
xu. With this modification to the states, the transient period can be reduced once
the IMP controller is switched on for fast convergence.
The adaptive IMP controller remains active as long as the error e(t) is excessive.
Once the error is insignificant, the value of f(t) starts to decrease as described by
f(t) = −Kd(t)f(t) (4.7)
Kd(t) =
Kdecay, |e(t)| < eu &∣∣e(t− T
4 )∣∣ < eu
0, otherwise
Since with a sinusoidal disturbance, the absolute error goes through zero every half
cycle, the insignificance of e(t) cannot be determined by condition |e(t)| < eu alone.
In addition, the error value at (t− T4 ) should also be compared with eu, where T = 2π
ω
is the estimated period of the sinusoid. If the error is less than the threshold at both
time instants, f(t) is then reduced exponentially at a rate of Kdecay. This smooth
removal of f(t) is adopted to avoid the phenomenon called chattering. Once f(t) is
reduced to insignificance, as measured by f l, it is set to zero. The IMP controller
stops the adaptation of the estimated frequency. With no input, a fixed sinusoid is
cancelled via an open loop mechanism. If perfect cancellation is not achieved, the
115
states of the monitored marginally stable internal model will grow linearly and will
eventually cause the switch S to close again. tl is set to current time t.
Table 4.1 lists reasonable methods of choosing related parameters for fast re-
sponse and maximum disturbance attenuation.
Table 4.1: Guidelines for parameter selections
K1, K2 Must be chosen such that the phase of the IMP controller plusthe phase of L(s) at ω is between −45◦ and −135◦.
eu Closed-loop level of noise and disturbance gives a reasonablevalue for eu.
xu eu value multiplied by the desired response time.
f l Between 0.05 and 0.2.Kdecay Can be chosen up to 5 times the reciprocal of the closed-loop
rise time.Kω Can be chosen that the dominant closed-loop poles are at least
twice faster than the frequency adaptation speed.
4.3 Switched System Model and Stability
Analysis
Let
x = Ax+Bu (4.8)
y = Cx (4.9)
116
be a state space representation of L(s) shown in Fig. 4.1, where x ∈ Rn. From (4.1)-
(4.2) and (4.8)-(4.9), by letting z = [xT , x1, x2]T , z ∈ R
n+2, the overall intermittent
control system shown in Fig. 4.1 can be expressed as
z =
A BK1 BK2
0 0 ω
−f(t)C −ω 0
z +
0
0
f(t)
r(t)
:= Aσ(t)z +Bσ(t)r(t) (4.10)
y =
[C 0 0
]z (4.11)
Based on the switching mechanisms described in Section 4.2, this intermittent
control system can be modelled as a switched system with two subsystems. The
switching signal σ(t) is defined as
σ(t) =
1, if f(t) ∈ (f l, 1]
2, if f(t) = 0
(4.12)
Let set {ti} represent the switching time instants with i being positive integer numbers
and ti ≤ ti+1. We also define pairs of subsets of {ti} for any q ∈ {1, 2} as follows
{tq,k} = {ti|when subsystem q is switched on},
{tq,k} = {ti|when subsystem q is switched off}
117
with tq,k < tq,k. Let S be a switching sequence associated with the switched system.
The interval completion I(S|q) is defined as the completion of the set of time intervals
during which subsystem q is active, i.e., I(S|q) =⋃k
[tq,k, tq,k
].
Matrices Aσ(t), Bσ(t) can be written as
A1(t) =
A BK1 BK2
0 0 ω
−exp(∫ tt1,k
−Kddτ )C −ω 0
, (4.13)
B1(t) =
0
0
exp(∫ tt1,k
−Kddτ)
, (4.14)
A2 =
A BK1 BK2
0 0 ω
0 −ω 0
, (4.15)
B2 =
0
0
0
, (4.16)
with t1,k ≤ t ≤ t1,k. Pair(A1(t), B1(t)
)describes the closed-loop control subsystem
with adaptive frequency estimation, and(A2, B2
)describes the open loop control
subsystem without frequency adaptation.
For simplicity, analysis of the algorithm has been performed with Ks set to
118
infinity. Under this condition, for the switched system described by (4.10) and (4.11),
we have
Theorem 4.1. If K1, K2, fl, and Kdecay are chosen such that the following as-
sumptions are satisfied,
1. The equilibrium of subsystem 1 is uniformly exponentially stable, i.e., there
exist finite positive constants γ1, λ1, such that ‖Φ1(t, τ)‖ ≤ γ1e−λ1(t−τ), where
Φ1(t, τ) is the transition matrix of subsystem 1;
2. There exists a constant µ2 > 0, such that each eigenvalue of A satisfies Re[eig(A)] ≤
−µ2;
3. The minimum active time of subsystem 1, τ1m, given by τ1m = − ln(f l)/Kdecay,
satisfies τ1m ≥ τs, where τs is dependent on Aσ(t),
the switched system (4.10), (4.11) is stable in the sense of Lyapunov.
The proof of this theorem is based on a multiple Lyapunov functions approach
developed in [11], and is similar to the proof of Theorem 1 in [10], thus is omitted
here. Note assumption 1 is guaranteed if the time derivative of matrix A1(t) is
upper bounded by the condition given in [12]. This condition is always satisfiable for
sufficiently small Kdecay.
119
4.4 Simulation Results
The performance of the intermittent cancellation control strategy is demonstrated via
two examples.
4.4.1 Example 1
A simple second order plant given by
xp =
−1 −1
1 0
xp +
1
0
u+
1 0
0 1
n(t) (4.17)
y =
[13 1
]xp (4.18)
is considered. As shown in Fig. 4.1, the disturbance consists of a sinusoidal signal,
r(t) = 1.5 sin(0.6t), and a vector of Gaussian noise n(t) with zero mean and standard
deviation 0.05. With this noise level, the standard deviation of the open loop system
output is 0.0037. The goal of this example is to design a wideband disturbance
controller and an adaptive internal model principle controller to identify the sinusoidal
frequency and minimize both disturbance components. This design is based on two
non-intermittent controllers which are introduced first.
120
4.4.1.1 Controller 1
This controller is a wideband disturbance controller and is designed using linear
quadratic regulator (LQR) method. The optimization criterion is defined as
J =
tf∫
ti
(y2 + uTRu)dt (4.19)
where R is a weighting matrix, ti, tf = ∞ are the initial and final times respectively.
The optimal state feedback gain is given by [13]
Km = −R−1BTp P, (4.20)
where P is the solution to the algebraic Riccati equation (ARE),
ATp P + PAp − P (BpR
−1BTp )P + CT
p Cp = 0 (4.21)
and (Ap, Bp, Cp) is a state representation of the plant, such as those given in (4.17)-
(4.18). By choosing the weighting matrixR = 0.2, we obtainKm = [−1.1106, −1.4495].
4.4.1.2 Controller 2
In order to reject the sinusoidal disturbance with unknown frequency, we need to tune
the adaptive IMP controller described by (4.1)-(4.3). Using the augmenting method
presented in [14], we can treat the two states of the IMP controller as additional
121
states together with the plant states to implement LQR method. The state equations
of this augmented system are
xp
x1
x2
= Ae
xp
x1
x2
+Beu+Wen(t) (4.22)
y = Ce
xp
x1
x2
(4.23)
where
Ae =
Ap 0 0
0 0 ω
−Cp −ω 0
, Be =
Bp
0
0
,
Ce =
[Cp 0 0
], We =
1 0
0 1
0 0
0 0
.
122
The optimization criterion (4.19) is then written as
J =
tf∫
ti
(y2 + ρ1x21 + ρ2x
22 + uTRu)dt (4.24)
for this augmented system. Note that weight must be applied to an augmented state
to satisfy the observability requirement of LQR control. The weighting R is the
same as in controller 1’s design. ρ1, ρ2 are two weightings on the IMP controller
states. For the choice of ρ1 = 0, ρ2 = 0.3, we obtain the optimal state feedback
gain for the augmented system as Ke = [−1.5708, −2.1427, −0.4159, 1.1520]. The
dominant closed-loop poles are−0.254±j0.539. The adaptation gain for the estimated
frequency is chosen as Kω = −0.1, such that the dominant poles are 2.5 times faster
than the adaptation speed.
4.4.1.3 Intermittent cancellation controller
In this example, L(s) in Fig. 4.1 is designed as the combination of the plant and
controller 1. Thus a state representation of L(s) is given by (Ap + BpKm, Bp, Cp),
with the same Km value as for controller 1. We take the same gain for the adaptive
IMP controller M(s) as in controller 2, which are K1 = −0.4159, K2 = 1.1520,
and Kω = −0.1. The standard deviation of the tracking error and the rise time
of the closed-loop system are measured as σ1 = 0.004 and tr = 25. According to
the guidelines in Table 4.1, corresponding parameters for the intermittent control are
chosen as f l = 0.1, eu = 0.01, xu = 0.25, Ks = 20, and Kdecay = 0.1. The initial
123
value of the estimated frequency is chosen as 0.5 rad/s.
The simulation was run for a total of 800 time units. The sinusoidal compo-
nent r(t) is only present between 199 and 612 while the Gaussian noise n(t) is always
present. To better demonstrate the disturbance rejection performance of the inter-
mittent control, we compare its optimization cost function values with controller 1
and controller 2. The cost functions are calculated using the following criterion
J =
tf∫
0
[y(t)− r(t)]2 + [u(t)− uss(t)]2Rdt (4.25)
where uss(t) is the steady state control signal, and tf is the simulation time. Since
the frequency response of the plant at ω = 0.6 rad/s is G(jω) = 1.1625∠ − 31.84◦,
in steady state, between t = 199 and t = 612, uss(t) = 1.2903 sin(0.6t + 31.84◦),
otherwise it is zero. For controller 1, uss(t) = 0.9794 sin(0.6t+173.47◦) for 199 < t <
612 and is zero otherwise. To distinguish between the transient response performance
and disturbance rejection capability of the approaches, the costs are broken into five
time intervals as follows.
(I) 0 ≤ t ≤ 199, only the additive Gaussian noise is present.
(II) 199 < t ≤ 250, transient period after the sinusoidal disturbance is introduced
at t = 199.
(III) 250 < t ≤ 612, steady state period after the sinusoid is introduced.
(IV) 612 < t ≤ 640, transient period after the sinusoid is removed at t = 612.
124
(V) 640 < t ≤ 800, steady state period after the sinusoid is removed.
The costs of the controllers are listed In Table 4.2 and Table 4.3. The inter-
mittent cancellation controller is referred to as controller 3. The costs of controller
2 and controller 3 in Table 4.2 are calculated with frequency adaptation. In Table
4.3, the costs are calculated when the frequency adaptation was turned off for both
controllers. In this case, the sinusoid’s frequency is known. Since there is no IMP
control in controller 1, the costs for both cases are the same.
Table 4.2: Costs of disturbance rejection performance with frequency adaptation
Interval Controller 1 Controller 2 Controller 3I 0.0034 0.0041 0.0034II 9.9529 13.9305 5.7661III 71.1713 0.0186 0.0174IV 0.0160 3.7715 3.7128V 0.0029 0.0043 0.0036
Table 4.3: Costs of disturbance rejection performance without frequency adaptation
Interval Controller 2 Controller 3I 0.0042 0.0034II 4.1699 3.6713III 0.0125 0.0115IV 3.7875 3.7462V 0.0039 0.0030
Compared to controller 2, controller 3 has 20% improvements during intervals
I and V. During transient interval II, controller 3 has a significant 1.4 times improve-
ment. The improvement of controller 3 during interval III is not significant, because
125
the algorithm has not completely converged. If the simulation time was extended, we
could get similar percentage improvements as during intervals I and V. From both
tables, we can see controller 3 has less costs than the other two controllers during each
time interval. Both controller 2 and controller 3 have less costs when the frequency
adaptation is turned off compared to unknown frequency case.
0 100 200 300 400 500 600 700 8000
0.2
0.4
0.6
0.8
1
Time
f(t)
Intermittent cancellation control action
0 100 200 300 400 500 600 700 800−0.5
0
0.5
1
1.5
Time
Tra
ckin
g er
ror
Figure 4.2: Intermittent IMP control and tracking error
The intermittent IMP control action is illustrated in the top plot of Fig. 4.2 by
showing the value of f(t) of the adaptive IMP controller. The tracking error e(t) is
shown in the bottom plot. The estimated disturbance frequency is shown in the top
plot of Fig. 4.3. Before the sinusoidal disturbance is introduced, the frequency stays
126
0 100 200 300 400 500 600 700 800
0.1
0.2
0.3
0.4
0.5
0.6
Time
ω (
rad/
s)
Frequency Estimation
Controller 3Controller 2
250 300 350 400 450 500 550 6000.598
0.599
0.6
0.601
0.602
Time
ω (
rad/
s)
Controller 3Controller 2
Figure 4.3: Frequency estimation of the intermittent and non-intermittent control
at its initial value 0.5 rad/s. When the monitored signal xm(t) is greater than the
threshold xu after the sinusoid is introduced at t = 199, the IMP controller is turned
on by setting f(t) = 1. The adaptation of the disturbance frequency starts. In the
mean time, the plant output starts to track the sinusoidal disturbance.
When the tracking error e(t) is less than eu at both the current instant and
one quarter period earlier as defined in section 4.2, f(t) starts to decay exponentially
with a rate of Kdecay. When f(t) decreases to the lower threshold f l, the input
to the adaptive IMP controller e(t) is removed by setting f(t) = 0. The estimated
frequency stops updating until next switch-on of the controller. This can be seen in
127
190 200 210 220 230 240 250
−1
−0.5
0
0.5
1
1.5
Time
Sin
usoi
d ad
ded
Transient tracking error
Controller 3Controller 2
610 615 620 625 630 635 640
0
0.5
1
Time
Sin
usoi
d re
mov
ed
Controller 3Controller 2
Figure 4.4: Tracking error during transient periods
the bottom plot of Fig. 4.3, where the estimated frequency using controller 2 is also
plotted for comparison. In Fig. 4.2, f(t) starts to decrease immediately after the
second switch-on of the controller which indicates the insignificance of the tracking
error. There remains an exponentially decaying residue error that continues to cause
the IMP controller to be invoked.
One of the stability requirements for controller 2 is that a predictable component
exists. However, this assumption is violated when t < 199 and t > 612. As a result,
controller 2’s frequency estimation changes drastically when t < 199 and t > 612,
which is shown in the top plot of Fig. 4.3. The abrupt removal of the sinusoidal
128
disturbance introduces a decaying exponential term in the output. Mathematically,
this can be modelled as a complex frequency with zero real part. The adaptive IMP
control algorithm ‘perceives’ this and responds by having the identified frequency
converge to zero. Controller 3 automatically avoids this by disabling the IMP control
when the predictable component is absent. As a result, the frequency stops adaptation
and stays as it is, which is shown in the top plot of Fig. 4.3. From this plot, it can
also be seen that estimated frequency of controller 2 decays towards zero, i.e. the
IMP controller becomes an integral controller. If the wideband disturbance controller
included in L(s) contains integral action, the two parallel integral controllers can lead
to stability problems. Controller 3 provides a major improvement to avoid this issue.
The tracking error of controller 3 during transient periods, during which the sinusoid
is added and removed, are shown in Fig. 4.4 together with controller 2’s performance
for comparison.
4.4.2 Example 2
In this example, a periodic disturbance with two sinusoidal components is considered.
That is
r(t) = 1.5 sin(0.6t) + sin(t+ 0.2)
The standard deviation of the zero mean Gaussian noise vector n(t) is 0.1, which
results in a standard deviation of 0.0071 for the open loop system output. The
129
same plant as in Example 1 is considered. The open loop system controller is
exactly the same as in previous example, which has the state feedback gain as
Km = [−1.1106,−1.4495].
4.4.2.1 Controller 2
Since there are two sinusoidal components in the disturbance, two corresponding
adaptive IMP controllers need to be designed for estimating both frequencies. Fol-
lowing the same augmentation method used in Example 1, a state representation of
the augmented system can be obtained as
xp
x11
x12
x21
x22
= Ae
xp
x11
x12
x21
x22
+Beu+Wen(t) (4.26)
y = Ce
xp
x11
x12
x21
x22
(4.27)
130
where x11, x12 and x21, x22 are the states of the two IMP controllers, ω1, ω2 are
estimates of the two sinusoids frequencies, and
Ae =
Ap 0 0 0 0
0 0 ω1 0 0
−Cp −ω1 0 0 0
0 0 0 0 ω2
−Cp 0 0 −ω 0
, Be =
Bp
0
0
0
0
,
Ce =
[Cp 0 0 0 0
], We =
1 0
0 1
0 0
0 0
0 0
0 0
.
The optimization criterion is
J =
tf∫
ti
(y2 + ρ11x211 + ρ12x
212 + ρ21x
221 + ρ22x
222 + uTRu)dt (4.28)
131
for this augmented system with corresponding weightings. By choosing ρ11 = ρ21 = 0,
ρ12 = 0.3, ρ22 = 0.2, the optimal state feedback gain is
Ke = [−1.8657,−2.6591,−0.6941,−1.0091,−0.0527,−0.9986]
4.4.2.2 Intermittent cancellation controller
For the two adaptive IMP controllers, their gains are the same as those of controller
2, which are K11 = −0.6941, K12 = −1.0091, K21 = −0.0527, K22 = −0.9986. The
initial estimates of the two frequencies are 1.2 rad/s and 0.8 rad/s respectively. The
frequency adaptation gain for both controllers is Kω = 0.05. The two IMP controllers
also share the same set of parameters for intermittent control, which are f l = 0.1,
xu = 0.25, eu = 0.01, Ks = 20, and Kdecay = 0.05.
The simulation was run for 800 time units for both controller 2 and the inter-
mittent cancellation controller (controller 3). The disturbance r(t) is present from
t = 98.37, and the Gaussian noise is present for the whole time. Using the same
criterion as in (4.25), cost functions are calculated over three time intervals: (I)
0 ≤ t ≤ 98.37, (II) 98.37 < t ≤ 200, and (III) 200 < t ≤ 800. The steady state
control signal for this example is calculated as
uss(t) = 1.2903 sin(0.6t+ 31.84◦) + 0.9487 sin(t+ 71.57◦)
132
for t > 98.37, otherwise it is zero. For controller 1, the steady state control signal is
uss(t) = 0.9794 sin(0.6t+ 173.47◦) + 0.7132 sin(t+ 161.94◦)
t > 98.37, otherwise it is zero.
Table 4.4 gives the costs of the three controllers’ rejection performance during
different time intervals. When the periodic disturbance r(t) is not present, controller
Table 4.4: Costs of rejection of multiple sinusoids
Interval Controller 1 Controller 2 Controller 3I 0.0053 0.0082 0.0053II 67.4366 11.7477 6.9143III 419.7938 2.2155 2.1941
3 has over 54% improvement than controller 2. Controller 3 has almost 70% less cost
than controller 2 during the transient after r(t) is introduced. After the transient dies
out, controller 3 still has less cost than that of controller 2.
Fig. 4.5 shows the intermittent control actions of the two IMP controllers,
where f1(t), f2(t) are the same functions as in (4.1) for the two IMP controllers,
respectively. The two estimated frequencies are shown in Fig. 4.6 . For comparison,
controller 2’s frequency estimates are also plotted. It can be seen that controller
2’s estimated frequencies decay towards zero when r(t) is absent, while controller 3’s
estimates stay at their initial values. The tracking errors of the two controllers during
the transient period are shown in Fig. 4.7.
133
0 100 200 300 400 500 600 700 8000
0.2
0.4
0.6
0.8
1
Time
f 1(t)
Intermittent IMP control action
0 100 200 300 400 500 600 700 8000
0.2
0.4
0.6
0.8
1
Time
f 2(t)
Figure 4.5: Intermittent IMP control action
4.5 Conclusions
A control strategy is presented for cancelling disturbances with both predictable and
unpredictable components. By intermittently connecting the input to an adaptive
IMP controller, which can cancel the predictable component (sinusoidal signal), the
control system operates between open and closed-loop control modes. In closed-loop
mode, the IMP controller can identify the unknown frequency of the sinusoid using
an adaptive algorithm. In open loop mode, a state feedback controller is imple-
134
0 100 200 300 400 500 600 700 8000.4
0.5
0.6
0.7
0.8
0.9
Time
ω1 (
rad/
s)
Frequency estimation
IntermittentNon−intermittent
0 100 200 300 400 500 600 700 8000.9
1
1.1
1.2
1.3
Time
ω2 (
rad/
s)
IntermittentNon−intermittent
Figure 4.6: Frequency estimation using adaptive IMP algorithm
mented to minimize the unpredictable component (white Gaussian noise). With the
IMP controller being turned off, the state feedback controller can be made more ag-
gressive while maintaining stability margin and control actions. The advantage of
the intermittent control strategy is demonstrated through numerical examples. This
intermittent control system is modelled as a switched system for stability analysis,
which is based on an extended multiple Lyapunov functions approach developed in
[11]. One of the main difficulties of safely implementing the algorithm in [7] is the re-
quirement of perfect knowledge of the number of sinusoids to be tracked or cancelled.
135
90 100 110 120 130 140 150 160 170 180 190 200−1.5
−1
−0.5
0
0.5
1
Time
Inte
rmitt
ent
Transient tracking error comparison
90 100 110 120 130 140 150 160 170 180 190 200−1.5
−1
−0.5
0
0.5
1
Time
Non
−In
term
itten
t
Figure 4.7: Transient tracking error comparison between intermittent and non-intermittent control
This switching algorithm can be used to eliminate that requirement.
Bibliography
[1] B. A. Francis and W. M. Wonham, “The Internal Model Principle of Control
Theory,” Automatica, vol. 12, pp. 457–465, 1976.
[2] L. A. Sievers and A. H. von Flotow, “Comparison and Extensions of Control
Methods for Narrow-band Disturbance Rejection,” IEEE Transactions on Signal
Processing, vol. 40, no. 10, pp. 2377–2391, Oct. 1992.
[3] J. R. Glover, “Adaptive Noise Canceling Applied to Sinusoidal Interferences,”
IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. ASSP-25,
no. 6, pp. 484–491, Dec. 1977.
[4] M. Bodson, J. S. Jensen, and S. C. Douglas, “Active Noise Control for Periodic
Disturbances,” IEEE Transactions on Control Systems Technology, vol. 9, no. 1,
pp. 200–205, Jan. 2001.
[5] P. A. Regalia, “An Improved Lattice-Based Adaptive IIR Notch Filter,” IEEE
Transactions on Signal Processing, vol. 39, no. 9, pp. 2124–2128, Sep. 1991.
[6] L. J. Brown and Q. Zhang, “Identification of Periodic Signals with Uncertain
Frequency,” IEEE Transactions on Signal Processing, vol. 51, no. 6, pp. 1538–
1545, Jun. 2003.
[7] L. J. Brown and Q. Zhang, “Periodic Disturbance Cancellation with Uncertain
Frequency,” Automatica, vol. 40, pp. 631–637, 2004.
[8] L. J. Brown, G. E. Gonye, and J. S. Schwaber, “Non-linear PI Control Inspired
by Biological Control Systems,” in Proceedings of the 37th IEEE Conference on
Decision and Control, Tampa, FL, Dec. 1998, pp. 1040–1045.
[9] L. J. Brown and J. S. Schwaber, “Intermittent Cancellation Control: A Control
Paradigm Inspired by Mammalian Blood Pressure Control,” in Proceedings of
the American Control Conference, San Diego, CA, Jun. 1999, pp. 139–143.
136
137
[10] J. Lu and L. J. Brown, “Stability Analysis of a Proportional with Intermittent
Integral Control System,” in Proceedings of the 2010 American Control Confer-
ence, Baltimore, MD, Jul. 2010, pp. 3257–3262.
[11] ——, “A Multiple Lyapunov Functions Approach for Stability of Switched Sys-
tems,” in Proceedings of the 2010 American Control Conference, Baltimore, MD,
Jul. 2010, pp. 3253–3256.
[12] C. A. Desoer, “Slowly Varying System x = A(t)x,” IEEE Transactions on Au-
tomatic Control, pp. 780–781, Dec. 1969.
[13] H. Kwakernaak and R. Sivan, Linear Optimal Control Systems. John Wiley
and Sons, Inc., 1972.
[14] C.-T. Chen, Linear System Theory And Design, 3rd ed. New York, NY: Oxford
University Press, 1999.
138
Chapter 5
Conclusions
5.1 Summary
Two advanced applications of the internal model principle control theory are discussed
in this thesis.
First, as a signal processing problem, an IMP based adaptive algorithm is devel-
oped for the identification of exponentially damped sinusoidal signals with unknown
parameters. Estimation of the two key parameter, damping factor and frequency, is
the focus of this algorithm. This algorithm is developed in discrete time based on
its continuous time version presented in [2]. A state space representation of the sig-
nal model in terms of estimated parameters is derived. The differences between the
parameter estimations and their true values can be described in terms of the model
state variables and the tracking error. By using integral controllers, the two param-
eter estimation errors can be eliminated asymptotically. Local exponential stability
and convergence of this adaptive algorithm is proved using a two time scale averaging
theory developed in [3].
139
Simulation results in the MATLAB/Simulink environment show that this algo-
rithm can not only identify constant parameters, but also track slowly time-varying
parameters of the EDS signal. By constructing a series of IMP controllers in parallel,
the adaptive feedback system can identify a signal composed of a sum of EDS modes,
with each IMP controller corresponding to one EDS mode. From the simulation re-
sults, this algorithm has shown its functionality despite the limitation that the slow
system shall be slower than the fast system. In order to speed up the algorithm, the
closed-loop poles can be placed closer to the origin by tuning the function L(z).
The second advance in IMP control theory is its application in switched systems.
For a system subjected to a disturbance with both predictable and unpredictable
components, two disturbance controllers need to be designed with their own attempts
to achieve best performance. An IMP controller is a perfect candidate for cancelling
the narrowband disturbance. A state feedback controller is designed using the LQR
method to minimize the white noise. But when both controllers are present in a
feedback system, they affect each other’s rejection capabilities. Especially, the state
feedback controller’s capability of minimizing the white noise is limited due to the
phase lag introduced by the IMP controller.
In order to achieve an optimal disturbance rejection performance, a combination
of open and closed-loop control strategy is presented. When the narrowband distur-
bance is significant, both controllers are active, in a closed-loop mode, to achieve fast
rejection. The system switches to open loop control mode when the narrowband dis-
140
turbance is cancelled within an acceptable range. In this control mode, the input to
the IMP controller, which is the tracking error, is disconnected. The IMP controller’s
output is maintained. Thus the state feedback controller is made more aggressive in
minimizing the white noise while maintaining the stability margins and control ac-
tions. Depending on the level of the tracking error, the IMP controller’s input may be
connected intermittently. Simulation results confirm the performance improvement of
this control strategy compared with two other controllers, a state feedback controller
and an augmented state feedback controller with IMP control action.
In this thesis, two types of narrowband signals are considered, constant and
periodic with unknown frequencies. For a constant disturbance, a simple IMP con-
troller, integrator, is used. For the periodic case, traditional adaptive IMP controllers
are used to first identify the unknown frequencies before the periodic disturbance
can be cancelled. If the periodic signal contains multiple sinusoidal components with
unknown frequencies, the same number of adaptive IMP controllers need to be placed
in parallel.
Due to its intermittent control fashion, this control strategy is modelled as a
switched system for stability analysis. Since existing stability analysis techniques
cannot be applied directly, an extended multiple Lyapunov functions approach is
developed to relax some constraints.
141
5.2 Concluding Remarks
Internal model principle control theory has drawn many researchers’ attention in the
area of output regulation. The adaptive IMP control algorithm developed in [1] is
capable of estimating frequencies of periodic or quasi-periodic signals to achieve zero-
error regulation. The advantages of applying this algorithm include its ability of
online real time fast estimation and low computational cost.
However, two of the main drawbacks of this approach is the requirement to
design stabilizing feedback gains and the need for a priori knowledge of the number
of signals to be cancelled. Note that the feedback design problem becomes increasingly
difficult and even infeasible as the number of frequencies increases.
By integrating the intermittent control to the adaptive IMP control algorithm,
the number of sinusoids to be cancelled is not necessarily required. Based on the
level of tracking error, the switching control mechanism determines the number of
adaptive IMP controllers to be turned on. This switching control thus makes the
feedback design feasible even with a large number of sinusoids.
This control strategy can also relax two design constraints in linear control,
which are the trade-off between response speed and noise rejection ability, as well
as the trade-off between the speed and actuator control effort. This improvement is
demonstrated through the simulation results in Chapter 3 and Chapter 4.
142
5.3 Future Work
Active acoustic noise control is an important area in many applications. The acoustic
noise can be cancelled using speakers. Due to the nature of some applications, such
as rotating electric fans, we know the noise signals are periodic but their frequencies
are uncertain. It will be of our interest to build a model of such application and
to implement the adaptive IMP control algorithm for identifying and cancelling the
noise.
From the simulation results in Chapter 2, we notice that the estimation per-
formance of the proposed adaptive algorithm degrades as the number of EDS modes
increases. It will be worthy investigating the estimation performance by combining
the intermittent control with the adaptive IMP control for EDS signals. And ap-
ply this approach to real musical signals, which have more complicated spectrum
contents. This would be a possible future work in this area.
Bibliography
[1] L. J. Brown and Q. Zhang, “Identification of Periodic Signals with Uncertain
Frequency,” IEEE Transactions on Signal Processing, vol. 51, no. 6, pp. 1538–
1545, Jun. 2003.
[2] J. Lu and L. J. Brown, “Internal model principle based control of exponentially
damped sinusoids,” International Journal of Adaptive Control and Signal Process-
ing, vol. 24, no. 3, pp. 219–232, 2010.
[3] E. Bai, L. Fu, and S. S. Sastry, “Averaging Analysis for Discrete Time and
Sampled Data Adaptive Systems,” IEEE Transactions on Circuits and Systems,
vol. 35, no. 2, pp. 137–148, Feb. 1988.
143
144
Curriculum Vitae
Name: Jin Lu
Post-secondary 2004-2006 M.E.Sc.Education and Electrical EngineeringDegrees: The University of Western Ontario
London, Ontario, Canada
1992-1996 B.Eng.Electrical EngineeringZhejiang UniversityHangzhou, China
Related Work Experience: 2004-2010 Research/Teaching AssistantThe University of Western OntarioLondon, Ontario, Canada
2000-2003 Product EngineerABB Beijing Drive Systems Co., Ltd.Beijing, China
1996-2000 Control Systems EngineerBeijing Mechanical Equipment InstituteBeijing, China
Publications:
[1] J. Lu and L. J. Brown, “Internal Model Principle Based Control of Exponen-tially Damped Sinusoids”, in International Journal of Adaptive Control and
Signal Processing, 24(2010), no.3, pp.219-232.
[2] J. Lu and L. J. Brown, “A Combination of Open and Closed-loop Controlfor Disturbance Rejection”, Submitted to 24th IEEE Canadian Conference on
Electrical and Computer Engineering, December, 2010.
145
[3] J. Lu and L. J. Brown, “Combining Open and Closed-Loop Control Via Inter-mittent Integral Control Action”, Submitted to IEEE Transactions on Auto-
matic Control, September, 2010.
[4] J. Lu and L. J. Brown, “A Multiple Lyapunov Functions Approach for Stabilityof Switched Systems” in Proceedings of the 2010 American Control Conference,Baltimore, MD, July, 2010, pp. 3253-3256.
[5] J. Lu and L. J. Brown, “Stability Analysis of a Proportional with IntermittentIntegral Control System” in Proceedings of the 2010 American Control Confer-
ence, Baltimore, MD, July, 2010, pp. 3257-3262.
[6] J. Lu and L. J. Brown, “Identification of Exponentially Damped SinusoidalSignals” in Proceedings of the 17th IFAC World Congress, Seoul, South Korea,July, 2008, pp. 5089-5094.
[7] J. Lu and L. J. Brown, “Control of Exponentially Damped Sinusoidal Signals”in Proceedings of the 2007 American Control Conference, New York City, NY,July, 2007, pp. 1937-1942.